CONIC SECTIONS
PARABOLA
1. CONIC SECTIONS:
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.
a) The fixed point is called the focus.
b) The fixed straight line is called the directrix .
c) The constant ratio is called the eccentrity denoted by e.
d) The line passing through the focus and perpendicular to the directrix is called the .
e) A point of intersection of a conic with its axis is called a vertex .
2. GENERAL EQUATION OF A CONIC: FOCAL DIRECTRIX PROPERTY:
The general equation of a conic with focus (p,q) and directs lx + my+ n= 0 is:
(l²+ m²)[(x - p)²+ (y - q)²]= e²(lx + my + n)²≡ ax²+ 2hxy + by²+ 2gx + 2fy + c=0
3. DISTINGUISHING BETWEEN THE CONIC:
The nature of the conic section depends upon the position of the focus S w.r.t. the directrix and also upon the value of the eccentricity e. Two different cases arise.
Case (i) : when the focus lies on the directrix:
In this case D ≡ ABC + 2 fgh - af²- bg²- ch²= 0 and the general equation of a conic represents a pair of straight lines and if:
e> 1 the lines will be real and distinct intersecting at S.
e= 1 the lines will coincident.
e< 1 the lines will be imaginary.
Case (ii): when the focus does not lie on the directrix :
The chronic represents :
A parabola : e= 1; D≠ 0; h½= ab
An ellipse : 0< e < 1; D≠ 0; h² < ab
A hyperbola: D≠ 0; e> 1; h² > ab
A rectangular hyperbola : e> 1; D ≠ 0 ; h² > ab ; a+ b = 0
PARABOLA
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed (focus) is equal to its perpendicular distance from a fixed line (directrix).
Standard equation of the parabola is y²= 4ax. For this parabola :
i) vertex is (0,0)
ii) Focus is (a,0)
iii) Axis is y= 0
iv) Directrix is x + a = 0
a) Focal distance:
The distance of a point in the parabola from the focus is called the focal distance of the point.
b) Focal chord:
A chord of the parabola, which passes through the focus is called a focal chord.
c) Double ordinate:
A chord of the parabola perpendicular to the axis of the symmetry is called double ordinate.
d) Latus rectum:
A double ordinate passing through the focus or a focal chord perpendicular to axis of parabola is called the latus rectum . For y²= 4ax.
• Length of the latus rectum= 4a.
• Length of the semi latus rectum= 2a.
• Ends of the latus rectum are L(a, 2a) and L'(a, -2a)
Note that:
i) Perpendicular distance from focus on directrix = half the latus rectum.
ii) Vertex is middle point of the focus and the point of intersection of the directrix and axis.
iii) Two parabolas are said to be equal if they have the same latus rectum.
5. PARAMETRIC REPRESENTATION:
The simple and the best form of representing the coordinates of a point on the parabola is (at², 2at). The equation x= at² and y= 2at together represents the parabola y²= 4ax, t being the parameter.
6. TYPE OF PARABOLA :
Four standard forms of the parabola are y²= 4ax; y²= - 4ax; x²= 4ay; x²= - 4ay
1) Find the vertex, axis, directrix, focus, latus rectum and the tangent at vertex for the parabola 9y²- 16x - 12y - 57= 0. (-61/16,2/3), y= 2/3, x= -613/144, (-485/144,2/3), 16/9, x= -61/16
2) The length of latus rectum of a parabola, whose focus is (2,3) and directrix is the line x - 4y +3=0 is
a) 7/√17 b) 14/√21 c) 7/√21 d) 14/√17. D
3) Find the equation of the parabola whose focus is (-6,-6) and vertex (-2,2). (2x - y)²+ 104x + 148y - 124 = 0
4) The extreme points of the latus rectum of a parabola are (7,5) and (7,3). Find the equation of the parabola. (y - 4)²= 2(x - 6.5) on x-axis at (14.5,0) , (y - 4)²= - 2(x - 7.5) on x-axis at (-0.5,0)
5) The name of the conic represented by the equation √(ax) + √(by)= 1, where a, b ∈ R, a, b > 0. Parabola
6) Find the vertex, axis, focus, directrix, latus rectum of the parabola 4y²+ 12x - 29y + 67=0. (-7/2,5/2); 5/2, (-17/4,5/2); x= -11/4, 3
7) Find the equation of the parabola whose focus is (1,-1) and whose vertex is (2,1). Also find its axis and latus rectum. 4x²+ y²- 4xy + 8x + 46y - 71= 0, 2x - y =3; 4√5
8) Find the equation of the parabola whose latus rectum is 4 units, axis is the line 3x + 4y= 4 and the tangent at the vertex is the line 4x - 3y +7=0. (3x + 4y -4)²= 20(4x - 3y +7)
POSITION OF A POINT RELATIVE TO A PARABOLA:
The point (x₁,y₁) lies outside, on or inside the Parabola y²= 4ax according as the expression y₁² - 4ax₁ is positive, zero or negative.
1) Find the value of α for which the point (α -1, α) lies inside the parabola y²= 4x. φ
CHORD JOINING TWO POINTS:
The angle of a chord of the Parabola y²= 4ax joining its two points P(t₁) and Q(t₂) is
y(t₁ + t₂) = 2x + 2at₁t₂
Note:
i) If PQ is focal chord then t₁t₂ = -1.
ii) Extremities of focal chord can be taken as (at², 2at) and (a/t², -2a/t)
2) Through the vertex O of a parabola y²= 4x chords OP and OQ are drawn at right angles to one another. Show that for all position of P, PQ cuts the axis of the parabola at a fixed point.
LINE & A PARABOLA
a) The line y= mx + c meets the Parabola y²= 4ax in two points real, coincide or imaginary according as a>=< cm => condition of tangency is, c= a/m.
Note: line y= mx + c will be tangent to parabola x²= 4ay if c= - am².
b) Length of the chord intercepted by the parabola y²= 4ax on the line y= mx + c is:
(4/m²) √{a(1+ m²)(a - mc)}.
Note: Length of the focal chord making an angle α with the x-axis is 4a cosec²α
7) If the line y= 3x + λ intersect the parabola y²= 4x at two distinct points then set of values λ is
a) (3, ∞) b) (-∞,1/3) c) (1/3,3) d) none. b
8) Find the value of a for which the point (a²-1,a) lies outside the parabola y²= 8x. (-∞. -√(8/7)) U (√(8/7, ∞)
9) The focal distance of a point on the parabola (x -1)²= 16(y -4) is 8. Find the coordinates. (7,8), (-9,8)
10) Show that the focal chord of parabola y²= 4ax makes an angle α with x-axis is of length 4a cosec²α.
11) Find the condition that the straight line ax + by + c=0 touches the parabola y²= 4kx. kb²= ac
12) Find the length of the chord of the parabola y²= 8x whose equation is x+ y =1. 8√3
LENGTH OF SUBTANGENT & SUBNORMAL:
PT and PG are the tangent and normal respectively at the point P to the parabola y²= 4ax. Then
TN= length of sub-tangent = twice the abscissa of the point P (Subtangent is always bisected by the vertex)
NG= length of subnormal which is constant for all points on the parabola and equal to its semilatus rectum (2a).
TANGENT TO THE PARABOLA y²= 4ax:
a) Point form:
Equation of tangent to the given parabola at its point (x₁, y₁) is yy₁ = 2a(x + x₁)
b) Slope form:
Equation of tangent to the given Parabola whose slope is 'm', is
y= mx + a/m, (m≠ 0)
Point of contact us (a/m², 2a/m)
c) Parametric form:
Equation of tangent to the given Parabola at its point P(t), is
ty= x + at²
Note: Point of intersection of the tangents at the point t₁ and t₁ is [at₁t₂, a(t₁+ t₂)], i.e., GM and AM of abscissa and ordinates of the points.
1) A tangent to the parabola y²= 8x makes an angle of 45° with the straight line y= 3x +5. Find its equation and its point of contact. y= x/2 + 4 point of contact (8,8)
2) Find the equation of the tangents to the parabola y²= 9x which go through the point (4,10). y= x/4 + 9 & y= 9x/4 +1
3) Find the locus of the point P from which tangents are drawn to the parabola y²= 4ax having slope m₁ and m₂ such that -
i) m₁+ m₂²= λ. y²- 2ax = λx²
ii) θ₁- θ₂= θ₀. y²- 4ax = (x + a)² tan²θ₀
Where θ₁ and θ₂ are the inclination of the tangents from positive x-axis.
4) Find the equation of the tangent to the parabola y²= 12x, which passes through the point (2,5). Find also the coordinates of their points of contact. x - y +3=0, (3,6); 3x - 2y +4=0, (4/3,4)
5) Find the equation of the tangents to the parabola y²= 16x, which are parallel and perpendicular respectively to the line 2x - y +5=0. Find also the coordinates of their points of contact. 2x - y + 2=0, (1,4); x + 2y + 16 =0, (16,-16)
6) Show that the locus of the point of intersection of tangents to the parabola y²= 4ax which meet at an angle θ is (x + a)² tan² θ = y²- 4ax.
NORMAL TO THE PARABOLA y²= 4ax:
a) Point form:
Equation of normal to the given parabola at its point (x₁, y₁) is
y - y₁ = - y₁(x - x₁)/2a
b) Slope form:
Equation of normal to the given Parabola whose slope is 'm', is
y= mx - 2am - am³
foot of the normal is (am², -2am)
c) Parametric form:
Equation of normal to the given parabola at its point P(t), is
y+ tx = 2at + at³
Note:
i) Point of intersection of normal at t₁ and t₁ is ({a(t₁²+ t₂²+ t₁t₂ +2), - at₁t₂(t₁ + t₂)]
ii) If the normal to the parabola at the point y²= 4ax at the point t₁, meets the parabola again at the point t₂.
then t₂ = -(t₁ + 2/t₁)
iii) if the normals to the parabola y²= 4ax at the points t₁ and t₂ intersect again on the parabola at point 't₃' then t₁t₂ = 2; t₃ = -(t₁, t₂) and the line joining t₁ and t₂ passes through a fixed point (-2a,0).
iv) If normal drawn to a parabola through a point P(h,k) then k= mh - 2am - am³, i.e., am³+ m(2a - h) + k = 0.
This gives m₁ + m₂ + m₃ =0; m₁m₂ + m₂m₃ + m₃m₁ = (2a - h)a; m₁m₂m₃= -k/a.
Where m₁, m₂, m₃ are the slopes of the three concurrent normals.
• Algebraic sum of slopes of the three concurrent normals is zero.
• Algebraic sum of ordinates of the three conormal points on the Parabola is 0.
• Centroid of the ∆ formed by three co-noormal points lies on the axis of parabola (x-axis).
1) Show that the normal to a parabola y²= 4ax at the point whose ordinate is equal to abscissa subtends a right angle at the locus.
2) If two normals drawn from any point to the parabola y²= 4ax make an angle α and β with the axis such that tanα . tanβ = 2, then find the locus of this point. y²= 4ax
3) Three normals are drawn from the point (14,7) to the curve y²- 16x - 8y = 0. Find coordinates of the feet of the normals. (0,0),(8,16),(3,-4)
4) If three distinct and real normals can be drawn to y²= 8x from the point (a,0), then
a) a> 2 b) a ∈ (2,4) c) a> 4 d) none c
5) Find the number of distinct normal that can be drawn from (-2,1) to the parabola y²- 4x - 2y -3=0. 1
6) If 2x + y + k =0 is a normal to the parabola y²= -16x, then find the value of k. 48
7) Three normals are drawn from the point (7,14)to the parabola x²- 8x -16y =0. Find the coordinates of the feet of the normals. (0,0),(-4,3),(16,8)
AN IMPORTANT CONCEPT
If a family of straight lines can be represented by an equation λ²P + λQ + R= 0 where λ is a parameter and P, Q, R are linear of x and y then the family of lines will be tangent to the curve Q²= 4PR.
1) If the equation m²(x +1) + m(y -2)= 0 represents a family of lines, where 'm' is parameter then find the equation of the curve to which these lines will always be tangents. y²= 4(x + y)
PAIR OF TANGENTS:
The equation of the pair of tangents which can be drawn any point P(x₁, y₁) outside the parabola to the Parabola y²= 4ax is given by: SS₁ = T² where:
S ≡ y² - 4ax; S₁ ≡ y₁²- 4ax₁ ; T≡ yy₁ - 2a(x + x₁).
DIRECTOR CIRCLE
Locus of the point of intersection of the perpendicular tangents to the Parabola y²= 4ax is called the director of circle. It's equation is x+ a = 0 which is Parabola 's own directrix.
1) The angle between the tangents drawn from a point (-a, 2a) to y²= 4ax is
a) π/4 b) π/2 c) π/3 d) π/6. b
2) The circle drawn with variable chord x + at -5=0 (a being a parameter) of the parabola y²= 20x as diameter will always touch the line
a) x +5=0 b) y +5=0 c) x + y+ 5=0 d) x - y +5=0 a
3) If the equation λ²x + λy - λ²+ 2λ +7=0 represents a family of lines, where λ is a parameter, then find the equation of the curve to which these lines will always be tangents. (y+ 2)²= 28(x -1)
4) Find the angle between the tangents drawn from the origin to the parabola, y²= 4a(x - a). π/2
CHORD OF CONTACT:
Equation of the chord of contact of tangents drawn from a point P(x₁, y₁) is
yy₁ = 2a(x + x₁)
Note: The area of the triangle formed by the tangents from the point (x₁, y₁) and the chord of contact is √(y₁²- 4ax₁)³⁾²/2a, also note that the chord of contact exists only if the point P is not inside.
1) If the line x - y -1=0 intersect the parabola y²= 8x at P & Q, then find the point of intersection of tangents at P and Q. (-1,4)
2) Find the locus of point whose chord of contact w.r.t to the parabola y²= 4bx is the tangent of the parabola y²= 4ax.
CHORD WITH A GIVEN MIDDLE POINT:
Equation of the chord of the Parabola y²= 4ax whose middle point is (x₁, y₁) is
y - y₁= 2a(x - x₁)/y₁.
This reduced to T= S₁, where T ≡ y y₁ -2a(x + x₁) and S₁ ≡ y₁²- 4ax₁
3) Find the locus of middle point of the chord of the parabola y²= 4ax which pass through a given (p,q). y²- 2ax - qy + 2ap=0
4) Find the locus of the middle of a chord of a parabola y²= 4ax which subtends a right angle at the vertex. y²= 2a(x - 4a)
5) Find the equation of the chord of contacts of tangents drawn from a point (2,1) to the parabola x²= 2y. 2x = y +1
6) Find the coordinates of the middle point of the chord of the parabola y²= 16x, the equation of which is 2x - 3y + 8=0. (14,12)
7) Find the locus of the midpoint of the chords of the parabola y²= 4ax such that tangent at the extremities of the chords are perpendicular. y²= 2a(x - a)
DIAMETER:
The locus of the middle points of a system of parallel chords of a Parabola is called a DIAMETER. Equation to the diameter of a Parabola is y= 2a/m, where m= slope of parallel chords.
IMPORTANT HIGHLIGHTS:
a) If the tangent and normal at any point P of the Parabola intersect the axis at T and G then ST= SG= SP where S is the focus. In other words the tangent and the normal at a point P on the Parabola are the bisectors of the angle between the focal radius SP and the perpendicular from P on the directrix. From this we conclude that all rays emanating from S will become parallel to the axis of the Parabola after reflection.
b) The portion of a tangent to a Parabola cut off between the directrix and the curve subtends a right angle at the focus.
c) The tangent at the extremities of a focal chord intersect at right angles on the directrix, and a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P(at², 2at) as diameter touches the tangent at the vertex and intercepts a chord of length a √(1+ t²) on a normal at the point P.
d) Any tangent to a Parabola and the perpendicular on it from the focus meet on the tangent at the vertex.
e) Semi latus rectum of the Parabola y²= 4ax, is the harmonic mean between segments of any focal chord of the Parabola is; 2a = 2bc/(b + c) i.e., 1/b + 1/c = 1/a
1) The parabola y²= 4x and x²= 4y divide the square region bounded by the line x=4, y= 4 and the coordinates axes. If S₁, S₂, S₃ are respectively the areas of these parts numbered from top to bottom then S₁: S₂: S₃. 1:1:1
2) Let P be the point (1,0) and Q a point on the parabola y²= 8x. The locus of the mid point of PQ. y²- 4x +2=0
Miscellaneous - 1
1) The common tangent of the parabola y²= 8ax and the circle x²+ y²= 2a² is
a) y= x+ a b) y+ x+ a =0 c) y+ x+ 2a =0 d) y= x+ 2a c
2) if the tangent to the parabola y²= 4ax meets the axis in T and tangent at the vertex A in Y and the rectangle TAYG is completed, show that the locus of G is y²+ ax=0.
3) If P(-3,2) is one end of the focal chord PQ of the parabola y²+ 4x + 4y = 0, then the slope of normal at Q is
a) -1/2 b) 2 c) 1/2 d) -2. a
4) Prove that the two parabolas y²= 4ax and y²= 4c(x - b) cannot have common normal, other than the axis unless b/(a - c) > 2.
5) If r₁, r₂ be the length of the perpendicular chords of the parabola y²= 4ax drawn through the vertex, then show that (r₁r₂)⁴⁾³ = 16a²(r₁²⁾³ + + r₂²⁾³).
6) if the tangents at P and Q meet in T, prove that
i) TP and TQ subtend equal angles of the focus S,
ii) ST²= SP.SQ,
iii) the triangle SPT and STQ are similar
7) The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.
8) Prove that the orthocentre of any triangle formed by three tangents to a parabola lies on the directrix .
BOOSTER - A
1) Latus rectum of the Parabola whose Focus is (3,4) and whose tangent at vertex has the equation x+ y = 7+ 5 √2 is
a) 5 b) 10 c) 20 d) 15
2) Directrix of a parabola is x+ y =2. If it's focus is origin, then latus rectum of the parabola is equal to
a) √2 units b) 2 units c) 2 √2 units d) 4 units
3) Which one of the following equations represents parametrically, parabolic profile ?
a) x=3 cos t; y= 4 sin t
b) x² -2 = - cos t; y= 4 cos²(t/2)
c) √x= tan t; √y= sec t
d) x= √(1- sin t); y= sin (t/2)+ cos(t/2)
4) Let C be a circle and L a line on the same plane such that C and L do not intersect . Let P be a moving point such that the circle drawn with centre at P to touch L also touches C. Then the locus of P is
a) a straight line parallel to L not intersecting C.
b) a circle concentric with C.
c) a parabola whose focus is centre of C and whose directrix is L
d) a Parabola whose focus is the centre of C and whose directrix is a straight line parallel to L.
5) If (t², 2t) is one end of a focal chord of the parabola y²= 4x then the length of the focal chord will be
a) (t + 1/t)² b) (t + 1/t) √(t²+ 1/t²)
c) (t - 1/t) √(t²+ 1/t²) d) none
6) From the locus of the parabola y²= 8x as centre, a circle is described so that a common chord of the curves is equidistance from the vertex and focus of the parabola. The equation of the circle is :
a) (x -2)²+ y²= 3
b) (x -2)²+ y²= 9
c) (x + 2)²+ y²= 9
d) x² + y² - 4x = 0
7) The point of intersection of the curves whose parametric equations are x= t²+ 1, y= 2t and x= 2s, y= 2/s is given by
a) (4,1) b) (2,2) c) (-2,4) d) (1,2)
8) If M is the foot of the perpendicular from a point P of a Parabola y²= 4ax to its directrix and SPM is an equilateral triangle, where S is the focus, then SP is equal to
a) a b) 2a c) 3a d) 4a
9) Through the vertex O of the Parabola y²= 4ax, variable chords OP and OQ are drawn at right angles. If the variable chord PQ intersects the axis of x at R, then distance OR:
a) varies with different position of P and Q.
b) equals to semi latus rectum of the parabola.
c) equals latus rectum of the Parabola.
d) equals double the latus rectum of the parabola.
10) The triangle PQR of area A is inscribed in the parabola y²= 4ax such that the vertex P lies at the vertex of the parabola and the base QR is a focal chord . The modulus of the difference of the ordinates of the points Q and R is
a) A/2a b) A/a c) 2A/a d) 4A/a
11) Point P lies on y²= 4ax and N is foot of perpendicular from P on its axis. A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q. NQ meets the tangent at the vertex in a point T such that AT = k NP, then the value of k is (where A is the vertex)
a) 3/2 b) 2/3 c) 1 d) none
12) The tangents to the parabola x= y²+ c from origin are perpendicular then c is equal to
a) 1/2 b) 1 c) 2 d) 1/4
13) The locus of a point such that two tangents drawn from it to the parabola y²= 4ax are such that the slope of one is double the other is
a) y²= 9ax/2 b) y²= 9ax/4 c) y²= 9ax d) x²= 4ay
14) T is a point on the tangent to a parabola y²= 4ax at its point P. TL and TN are the perpendicular on the focal radius SP and the directrix of the parabola respectively. Then
a) SL= 2(TN) b) 3(SL) = 2(TN) c) SL= TN d) 2(SL) = 3(TN)
15) The equation of the circle drawn with the focus of the parabola (x -1)² - 8y =0 as its centre and touching the parabola at its vertex is
a) x²+ y²- 4y =0
b) x²+ y²- 4y +1 =0
c) x²+ y²- 2x - 4y =0
d) x²+ y²- 2x - 4y +1 =0
16) Length of the normal chord of the parabola , y²= 4x, which makes an angle of π/4 with the axis of x is -
a) 8 b) 8√2 c) 4 d) 4√2
17) Tangents are drawn from the point (-1,2) on the parabola y²= 4x. The length, these tangents will intercept on the line x=2.
a) 6 b) 6√2 c) 2√6 d) none
18) Locus of the point of the intersection of the perpendiculars tangent of the curve y² + 4y - 6x + 4y -2 =0 is
a) 2x -1= 0 b) 2x + 3= 0 c) 2y +3 = 0 d) 2x + 5 = 0
19) Tangents are drawn from the points on the line x - y + 3= 0 to parabola y²= 8x. Then the variable chords of contact pass through a fixed point whose coordinates are
a) (3,2) b) (2,4) c) (3,4) d) (4,1)
20) The line 4x -7y + 10 = 0 intersects the parabola, y²= 4x at the point A and B. The coordinates of the point of intersection of the tangent drawn at the point A and B are
a) (7/2, 5/2) b)(- 5/2, 7/2) c) (5/2, 7/2) d) (- 7/2, 5/2)
21) From the point (4,6) a pair of tangents lines are drawn to the parabola y²= 8x. The area of the triangle formed by these pair of tangent lines and the chord of contact of the point (4,6) is
a) 2 b) 4 c) 8 d) none
22) TP and TQ are tangents to the parabola, y²= 4ax at P and Q. If the chord PQ passes through the fixed point (-a, b) then the locus of T is
a) ay= 2b(x - b) b) bx = 2a(y - a) c) by= 2a(x - a) d) ax = 2b(y - b)
23) If the tangent of the point P(x₁, y₁) to the parabola y²= 4ax meets the parabola y²= 4a (x + b) at Q and R, then the midpoint of QR is
a) (x₁+ b, y₁+ b)
b) (x₁- b, y₁- b)
c) (x₁, y₁) d) (x₁+ b, y₁)
24) Let PSQ be the focal chord of the parabola y²= 8x. if the length of SP= 6 then, l(SP) is equals to (where S is the focus)
a) 3 b) 4 c) 6 d) none
25) Two parabolas y²= 4a(x - l₁) and x¹= 4a(y - l₂) always touch one another, the quantities l₁ and l₂ are both variable. Locus of their point of contact has the equation.
a) xy= a² b) xy= 2a² c) xy= 4a² d) none
26) Equation x²- 2x - 2y +5=0 represents
a) a parabola with vertex (1,2)
b) a parabola with vertex (2,1).
c) a parable with directrix y= 3/2
d) a Parabola with directrix, y= 2/5
27) The normals to the parabola y²= 4ax from the point (5a, 2a) are
a) y= -3x + 33a b) x = -3y + 3a c) y= x - 3a d) y= -2x + 12a
28) The equation of the lines joining the vertex of the parabola y²= 6x to the points on it whose abscissa is 24, is
a) 2y + x + 1= 0 b) 2y - x + 1= 0 c) 2y + x = 0 d) - 2y + x = 0
29) The equation of the tangent to the parabola y²= 9x which passes through the point (4, 10) is
a) 4y + x + 1=0 b) - 4y + x + 36=0 c) 4y + 9x + 4= 0 d) 4y + 9x - 4= 0
30) Consider the equation of a parabola y²= 4ax, (a< 0) which of the following is false
a) tangent at the vertex is x= 0
b) directrix of the parabola is x= 0
c) vertex of the parabola is at the origin
d) focus of the parabola is at (a,0)
1c 2c 3b 4d 5a 6b 7b 8d 9c 10c 11b 12d 13a 14c 15d 16b 17b 18d 19c 20c 21a 22c 23c 24a 25c 26ac 27cd 28cd 29bc 30bd
CHECK THE STATUS
BOOSTER - B
1) The straight line joining any point P on the parabola y²= 4ax to the vertex and perpendicular from the focus to the tangent at P, intersect at R, then the equation of the locus of R is
a) x²+ 2y²- ax =0
b) 2x²+ 2y²- 2ax =0
c) 2x²+ 2y²- ay =0
d) 2x²+ 2py²- 2ay =0
2) Let A be the vertex and L the length of the lotus lactom of the parabola y² -2y - 4x -7 =0. The equation of the parabola with point A as vertex, 2L as the length of the latus rectum and the axis at right angles to that of given curve is
a) x²+ 4x + 8y - 4 =0
b) x²+ 4x - 8y +12 =0
c) x²+ 4x + 8y +12 =0
d) x²+ 8x - 4y + 8 =0
3) The Parametric coordinates of any point of the parabola y² = 4ax can be-
a) (at²,2at) b) (at²,-2at) c) (a sin²t, 2a sin t) d) (a sin t, 2a cos t)
4) PQ is a normal chord of the parabola y²= 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR is
a) equal to the length of the lauts rectum
b) equal to the focal distance of the point P.
c) equals to twice the focal distance of the point P.
d) equal to the distance of the point P from the directrix .
5) The length of the chord of the parabola y²= x which is bisected at the point (2,1) is
a) 5√2 b) 4√5 c) 4√50 d) 2√5
6) if the tangents and normal at the extmities of a focal chord of a parabola intercept at (x₁, y₁) and (x₂, y₂) respectively , then.
a) x₁= x₂ b) x₁= y₂ c) y₁= y₂ d) y₁= x₂
7) Locus of the intersection of the tangents at the ends of the normal chords of the parabola y²= 4ax is
a) (2a + x)y²+ 4a³= 0
b) (2a + x)y²+ 4a² = 0
c) (2a + y)x²+ 4a³= 0
d) none
8) The locus of the midpoint of the focal radii of a variable point moving on the parabola y²= 4ax is a parabola whose
a) latus rectum is half the latus rectum of the origin parabola.
b) vertex is (a/2,0)
c) directrix is y-axis
d) focus has the coordinates (a,0)
9) The equation of a straight line passing through the point (3,6) and cutting the curve y= √x orthogonally is
a) 4x + y - 18=0 b) x + y - 9=0 c) 4x - y - 6 =0 d) none
10) The tangent and normal for all real positive t, to the parabola y²= 4ax meet the axis of the parabola in T and G respectively, then the angle at which the tangent at P to the parabola is inclined to the tangent at P to the circle through the points P, T and G is
a) cot⁻¹ t b) cot⁻¹ t² c) tan⁻¹ t d) sin⁻¹{t/√(1+ t²)}
11) A variable circle is described to passes through the point (1,0) and tangent to the curve y= tan(tan⁻¹x). The locus of the centre of the circle is a parabola whose-
a) length of the latus rectum is 2√2.
b) axis of symmetry has the equation x+ y =1
c) vertex has the coordinates (3/4,1/4). d) none
12) AB, AC are tangents to a parabola y²= 4ax. p₁ p₂ and p₃ are the lengths of the perpendiculars form A, B and C respectively on any tangent to the curve, p₁ p₂ and p₃ are in
a) AP b) GP c) HP d) none
13) Through the vertex O of the parabola, y²= 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If θ₁, θ₂ and φ are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotθ₁ + cotθ₂ =
a) - 2 tanφ b) - 2 tan(π - φ) c) 0 d) 2 cotφ
14) Two parabolas have the same focus. If their directrix are the x-axis and y axis respectively, then the slope of their common chord is.
a) 1 b) -1 c) 4/3 d) 3/4
15) Tangent to the parabola y²= 4ax at point P meets the tangent at vertex A, at point B and the axis of parabola at T. Q is any point on this tangent and N is the foot of perpendicular from Q on SP, where S is focus. M is the foot of perpendicular from Q on the directrix then -
a) B bisects PT b) B trisects PT c) QM= SN d) QM= 2SN
16) If the distance between a tangent to the parabola y²= 4x and and a parallel normal to the same parabola is 2√2, then possible values of gradient of either of them are:
a) -1 b) +1 c) - √(√5 -2) d) + √{√5 -2}
17) if two distinct chords of a parabola x²= 4ay passing through (2a, a) are bisected on the line x+ y = 1, then the length of the latus rectum can be-
a) 2 b) 1 c) 4 d) 5
18) If PQ is a chord of parabola x²= 4y which subtends right angle at vertex. Then locus of centroid of triangle PSQ (S is a focus) is a parabola whose -
a) vertex is (0,3) b) length of LRA is 4/3 c) axis is x= 0 d) tangent at the vertex is x= 3
19) Identify the correct statement/s-
a) In a parabola vertex is the midpoint of focus and foot of the directrix .
b) P(at₁², 2at₁) & Q(at₂², 2at₂) are two points on y²= 4ax such that t₁t₁ = -1, then normal at P and Q are perpendicular.
c) There does not exist any tangent of y²= 4ax which is parallel to x-axis .
d) At most two normal can be drawn to a parabola from any point on its plane.
20) For parabola y²= 4ax consider three points A, B, C lying on it. If the centroid of ∆ ABC is (h₁, k₁) & centroid of triangle formed by the point of intersection of tangents at A, B, C has coordinates (h₂, k₂), then which of the following is always true -
a) 2k₁ = k₂ b) k₁ = k₂ c) k₁² (4a/3) (h₁ + 2h₂ ) k₁² = (4a/3) (2h₁ + h₂).
1b 2ab 3ab 4c 5d 6c 7a 8abcd 9a 10cd 11bc 12b 13a 14ab 15ac 16abcd 17ab 18abc 19abc 20bc
BOOSTER - C
Column - I
A) The normal chord at a point on the parabola y²= 4x substend a right angle at the vertex, then t² is
B) The area of the triangle inscribed in the curve y²= 4x. if the parameter of the vertices are 1, 2 and 4 is
C) The number of distinct normal possible from (11/4, 1/4) to the parabola y²= 4x is
D) The normal at (a, 2a) on y²= 4ax meets the curve again at (at², 2at), then the value of |t -1 | is
Column II
p) 4
q) 2
r) 3
s) 6
Aq Bs Cq Dp
BOOSTER - D
Column I
A) Area of a Triangle formed by the tangents drawn from a point (-2,2) to the parabola y²=4(x + y) and their corresponding chord of contact is
B) Length of the latus rectum of the conic 25[(x -2)²+ (y -3)²]= (3x + 4y - 6)² is
C) If focal distance of a point on the parabola y= x²- 4 is 25/4 and points are of the form (±√a, b) then value of a+ b is
D) Length of side of an equilateral triangle inscribed in a parabola y²- 2x - 2y -3=0 whose one angular point is vertex of the parabola, is
Column II
p) 8
q) 4√3
r) 4
s) 24/5
Ar Bs Cp Dq
BOOSTER - E
ASSERTION & REASON
These questions contain , Statement- I (assertion ) and Statement- II (reason)
A) Statement- I is true, Statement - II is true: Statement- II is correct explanation for Statement- I
B) Statement- I is true, Statement- II is true: Statement- II is NOT a correct explanation for Statement- I.
C) Statement- I is true, Statement- II is false.
D) Statement- I is false, Statement- II is true.
1) Statement- I: If normal at the ends of double ordinate x= 4 of parabola y²= 4x meet the curve again at P and P' respectively, because PP'= 12 unit .
Because
Statement II: If normal at t₁ of y²= 4ax meet the parabola again at t₁²= 2+ t₁t₂
a) A b) B c) C D) D
2) Statement - I: The lines from the vertex to the two extremities of a focal chord of the parabola y²= 4ax are at an angle of π/2.
Because
Statement II: If extrinities of focal chord of parabola are (at₁², 2at₁) and (at₂², 2at₂), then t₁t₂= -1
a) A b) B C) C D) D
3) Statement - I: If P₁Q₁ and Q₂Q₂ are two focal cords of the parabola y²= 4ax, then the locus of point of intersection of chords P₁P₂ and Q₁Q₂ is directrix of the parabola. Here P₁P₂ and Q₁Q₂ are not parallel.
Because
Statement - II: The focus of point of intersection of perpendicular tangents of parabola is directrix of the parabola.
1c 2d 3b
BOOSTER - F
COMPREHENSION BASED QUESTIONS
Comprehension
Observe the following facts for a parabola:
i) Axis of the parabola is only line which can be perpendicular bisector of the two chords of the parabola.
ii) If AB and CD are two parallel chords of the parabola and the normals at A and B intersect at P and the the normals at C and D intersect at Q, then PQ is a normal to the parabola.
Let a parabola is passing through (0,1), (-1,3),(3,3) & (2,1)
On the basis of above information, answer the following questions:
1) The vertex of the parabola is-
a) (1, 1/3) b) (1/3,1) c) (1,3) d) (3,1)
2) The directrix of the parabola is-
a) y- 1/24= 0 b) y + 1/2 = 0 c) y + 1/24= 0 d) y + 1/12= 0
3) For the parabola y²= 4x, AB and CD are any parallel chords having slope 1, C₁ is a circle passing O, A and B and C₂ is a circle passing through O, C and D. C₁ and C₂ intersect at.
a) (4,-4) b) (-4,4) c) (4,4) d) (-4,-4)
1a 2c 3a
BOOSTER - G
Comprehension:
If a source of light is placed at the fixed point of a parabola and if the parabola is a reflecting surface, then the ray will bounce back in a line parallel to the axis of the parabola.
On the basis of above information, answer the following questions:
1) A ray of light is coming along the line y= 2 from the positive direction of x-axis in strikes a concave mirror whose intersection with xy-plane is a Parabola y²= 8x, then the equation of the reflected ray is -
a) 2x + 5y = 4 b) 3x + 2y = 6 c) 4x + 3y = 8 d) 5x + 4y = 10
2) A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is y²+ 10y - 4x +17=0. After reflection, The ray must pass through the point -
a) (-2,-5) b) (-1,-5) c) (-3,-5) d) (-4,-5)
3) Two rays of light coming along the line y= 1 and y= -2 from the positive direction of x-axis and strikes a concave mirror whose intersection with the xy-plane is a parabola y²= x at A and B respectively. The reflected rays passes through a fixed point C, then the area of the triangle ABC is-
a) 21/8 sq.unit b) 19/2sq.unit c) 17/2sq.unit d) 15/2 sq.unit
1c 2b 3a
BOOSTER - H
1) Find the equation of parabola, whose focus is (-3,0) and a directrix is x +5=0.
2) Find the vertex, axis , focus, directrix, latus rectum of the parabola x²+ 2y - 3x + 5 = 0.
3) Find the equation of the parabola whose focus is (1,-1) and whose vertex is (2,1). Also find its Axis and latus rectum.
4) if the end point P(t₁) and Q(t₂) of a chord of a parabola y²= 4ax satisfy the relation t₁t₂ = k (constant) then prove that the chord always passes through a fixed point. Find that point also ?
5) Find the locus of the middle points of all chords of the parabola y²= 4ax which are drawn through the vertex.
6) O is the vertex of the parabola y² = 4ax & L is the upper end of the latus rectum. If LH is drawn perpendicular to OL meeting OX in H, prove that the length of the double coordinate through H is 4a √5.
7) Find the length of the side of an equilateral triangle inscribed in the parabola, y²= 4x so that one of its angular point is at the vertex.
8) Two perpendicular chords are drawn from the origin O to the parabola y= x², which meet the parabola at P and Q. Rectangle POQR is completed. Find the locus of vertex R.
9) Find the set of values of α in the interval [π/2, 3π/2] for which the point (sinα, cosα) does not lie outside the parabola 2y²+ x - 2=0.
10) Find the length of the focal chord of the Parabola y²= 4ax whose distance from the vertex is p.
11) If 'm' varies then find the range of c for which the line y= mxn+ c touches the parabola y²= 8(x +2).
12) Find the equations of the tangents to the parabola y²= 16x, which are parallel and perpendicular respectively to the line 2x - y +5=0. Find also the coordinates of their points of contact.
13) Find the equations of the tangents of the parabola y²= 12x which passes through the point (2, 5).
14) Prove the locus of the middle points of all tangents drawn from points on the directrix to the parabola y²= 4ax is y²(2x + a)= a(3x + a)².
15) Two tangent to the parabola y²= 8x meet the tangent at its vertex in the points P and Q. If PQ = 4 units, prove that the locus of the point of the intersection of the two tangents is y²= 8(x +2).
16) Find the equation of the circle is passes through the focus of the parabola x²= 4y and touches is it at the point (6,9).
17) In the parabola y²= 4ax, the tangent at the point P, whose abscissa is equal to the latus rectum meets the axis in T and the normal at P cuts the Parabola in Q. Prove that PT : PQ= 4:5.
18) Show that the normal at the points (4a, 4a) and at the upper end of the latis rectum of the parabola y²= 4ax intersect of the same Parabola .
19) Show that the locus of a point, such that two of the three normals drawn from it to the parabola y²= 4ax are perpendicular is y²= a(x - 3a).
20) if the normal at P(18,12) to the parabola y²= 8x cut it again at Q, show that 9PQ = 80√10
21) Prove that the locus of the middle point of portion of a normal to y²= 4ax intercepted between the curve and the axis is another parabola. Find the vertex and the latus rectum of the second Parabola.
22) A variable chord PQ of the parabola y²= 4x is drawn parallel to the line y= x. If the parameters of the points P and Q on the parabola are p and q respectively. Show that p+ q= 2. Also show that so that the ocus of the point of intersection of the normals at P and Q is 2x - y = 12.
23) P and Q are the points of contact of the tangents drawn from the point T to the parabola y²= 4ax. If PQ be the normal to the parabola at P. Prove that TP is bisected by the directrix .
24) The normal at a point P to the parabola y²= 4ax meets its axis at G. Q is another point on the parabola such that QG is perpendicular to the axis of the parabola. Prove that QG²- PG²= constant.
25) Three normals to y²= 4x pass through the point (15, 12). Show that one of the normals is given y= x - 3 and find the equations of the others.
1) y²= 4(x +4) 2) (3/2,-11/8), (3/2,-15/8), x= 3/2, y= -7/8, 2
3) (2x - y -3)²= -20(x + 2y -4), 2x - y -3=0, 4√5
4) (-k, 0) 5) y²= 2ax 7) 8√3 8) x²= y -2
9) a belongs to {π/2, 5π/6)U [π, 3π/2]
10) 4a³/p²
11) (-∞,-4) U [4, ∞)
12) 2x - y +2=0; (1,4), x + 2y +16 =0;(16,-16)
13) 3x - 2y + 4=0; x - y +3=0
16) x² + y² + 18x - 28y + 27 =0;
21) (a,0); a
25) y= -4x +72; 3x - y 33 =0;
BOOSTER - I
1) If from the vertex of a parabola a pair chords be drawn at right angles to another. And these chords as adjacent sides a rectangle be constructed, then find the locus of the outer corner of the rectangle.
2) Two perpendicular straight lines through the locus of the parabola y²= 4ax meets its directrix in T and T' respectively. Show that the tangents to the parabola parallel to the perpendicular lines intersect in the midpoint of TT'.
3) Find the condition on 'a' and 'b' so that the two Tangents drawn to the parabola y²= 4ax from a point are normal to the parabola x²= 4by.
4) TP and TQ are tangents to the parabola and the normal at P and Q meet at a point R on the curve. Prove that the centre of the circle circumscribing the triangle TPQ lies on the Parabola 2y²= a(x - a).
5) Let S is the focus of the parabola y²= 4ax and X the foot of the directrix . PP' is a double ordinate of the curve and PX meets the curve again in Q. Prove that P'Q passes through the focus .
6) Prove that on Axis of any parabola y²= 4ax there is a certain point K which has the property that, if a chord PQ of the parabola be drawn through it, then 1/(PK)² + 1/(QK)² is same for all positions of the chord. Find also the coordinates of the point K.
7) If (x₁, y₁), (x₂, y₂) and (x₃, y₃) be three points on the parabola y²= 4ax and the normals at these points meet in a point, then prove that (x₁ - x₂)/y₃ + (x₂ - y₃/y₁ + (x₃ - x₁)/y₂ = 0
8) A variable chord joining the points P((t₁) and Q(t₂) of the parabola y²= 4ax subtends a right angle at a fixed point t₀ of the curve. Show that it passes through a fixed point. Also find the co-ordinate of the fixed point.
9) Show that a circle circumscribing the Triangle formed by three co-normal points passes through the vertex of the parabola and its equation, 2(x²+ y²) - 2(h + 2a)x - ky = 0, where (h,k) is the point from where three concurrent normals are drawn.
10) A ray of light is coming along the line y= b from the positive direction of x-axis and strikes a concave mirror whose intersection with the xy-plane is a Parabola y²= 4ax. Find the equation of the reflected ray and show that it passes through the focus of the parabola. Both a and b are positive.
1) y²= 4a(x - 8a). 3) a²> 8b². 6) (2a,0) 8) [a(t₀²+4), 2at₀] 10) 4abx + (4a²- b²) - 4a²b = 0
BOOSTER- J
1) The angle between a pair of tangents drawn from a point P to the parabola y²= 4ax is 45°. Show that the locus of the point P is a hyperbola.
2) If the line x -1=0 is the directors of the parabola y⁴- Kx +8=0, then one of the values of k is
a) 1/8 b) 8 c) 4 d) 1/4
3)a) The equation of the common tangent touching the circle (x -3)²+ y²= 9 and the parabola the x-axis is
a) √3 y = 3x +1 b) √3 y = -(x +3) c) √3 y = x +3 d) √3 y = -(3x +1)
b) The equation of the directrix of the parabola y²+ 4y + 4x +2=0 is
a) x= -1 b) x= 1 c) x = -3/2 d) x= 3/2
4) The locus of the midpoint of the line segment joining the focus to a moving point on the parabola y²= 4ax another Parabola with directrix.
x= x= - a b) x= -a/2 c) x= 0 d) x= a/2
5) The equation of the common tangent to the curves y²= 8x and xy= -1 is
a) 3y= 9x +2 b) y= 2x +1 c) 2y= x +8 d) y= x +2
6) is a tangent to the circle then the set of possible values of the slope of the squads are normal to the parabola if focus is a part of the parabola itself to tangents are drawn from the point 40 to the parable angles between the tangent is at any point on the parabola a tangent is drawn which the directory find the locus appoint which channel in the ratio tangent to the curve at point 17 touching the circle at a point then coordinates of the access of a parabola is along the line in the distance of a vertex from origin is and that of the origin from spokens is 22 a vertex in the focus both the line the first quarter in the position of the parabola is the equation of the common tangent with the parabola are
Normal donate appoint lying in a parabola which intersects of 30 then area of the pqr radius of the circumcircle centroid or circumcenter of
β μ λ φ ∞ α ₁₂₃₄ ₀ θ α and β ∈ ₁₁₂₂ cot⁻¹ t ₃₄₅₆
α and β ∈ ₁₁₂₂ cot⁻¹ t ₃₄₅₆
β μ λ φ ∞ α ₁₂₃₄ ₀ θ
Square of the sad length to units is the circle to the vertices in the circle touching all the sides of the square is a line through if is a point and is another point is equals to 0.751.2510.5 a circle touches the line and external is such that both the circles are on the same of the line when the locus of the centre of the circle airlines through is drawn parallel point most such that is distance from the line and the vertex are equally low cost and then area of circle and the parabola the intersect at in the first on the fourth quarter in respectively tangent to the circle intercept at the x-axis and tangent to the parabola interceptor the exacts is at the ratio of the areas of the tangents the radius of the circumcircle of the triangle to 333 23 the radius of the encircle of the triangle for 3832 the curve is symmetry to the respect to line because a parabola symmetric about to success chachi Chadar only at 1 point touch Each other exactly at 2 points intersect but not touch at the exactly 2 points neither is per not touch each other the tangent in the normal to the parabola at a point on its needs its Axis at the point respectively the locus of the centroid of the triangle is a parabola is vertex directrix network
ELLIPSE
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