CONICS
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 eccentricity denoted by e.
d) The line passing through the focus & perpendicular to the directrix is called the axis.
e) A point of intersection of a conic with its axis is called vertex.
2. General equation of a conic: Focal Directrix Property:
The general equation of a conic with focus (p,q) & directrix lx + my + n=0 is
(l²+ m²)[(x - p)²+ (y - q)²]= e²(lx + my + n)² ≡ ax¹+ 2hxy+ by²+ 2gx + 2fy + c=0
3. Distinguish 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 lie on the directrix:
In this case D≡ abc+ 2fgh - af²- bg² - ch²=0 and the general equation of a conic represent a pair of straight lines and if:
e> 1 the line will be real and distinct intersecting at S.
e= 1 The lines will be coincident.
e < 1 The lines will be imaginary.
Case(ii): When the focus does not lie on the directrix:
e=1; D≠ 0 and h²= ab a parabola
0< e < 1; D≠ 0 and h²< ab an ellipse
D≠ 0; e> 1 and h²> ab. a hyperbola
e> 1; D≠ 0 and h²> ab; a+ b = 0 . A rectangular hyperbola
4. PARABOLA
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix).
Standard equation of a 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 on 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 a double ordinate.
d) Latus rectum:
A double ordinate passing through the focus of a focal chord perpendicular to the 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= L(a, 2a) & L'(a, -2a)
Notes 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 directrix and axis.
iii) Two parabola are said to be equal if they have the same latus rectum.
5. Parametric Representation:
The simplest and the best form of representing the coordinates of a point on the parabola is (at², 2at). The equation x= at² & y= 2at together represents the parabola y²= 4ax, t being the parameter.
6. Type of PARABOLA
Four standards forms of the parabola are y²= 4ax; y²=- 4ax; x²= 4ay; x²= -4ay
When parabola y²= 4ax
Vertex: (0,0)
Focus: (a,0)
Axis: y= 0
Directrix: x= - a
Length of latus rectum: 4a
Ends of Latus rectum: +a, ±2a)
Parametric equation: (at²,2at)
Focal length: x + a
When parabola y²= - 4ax
Vertex: (0,0)
Focus: (- a,0)
Axis: y= 0
Directrix: x= a
Length of latus rectum: 4a
Ends of Latus rectum: (-a, ±2a)
Parametric equation: (-at²,2at)
Focal length: x - a
When parabola x²= 4ay
Vertex: (0,0)
Focus: (0,a)
Axis: x = 0
Directrix: y= - a
Length of latus rectum: 4a
Ends of Latus rectum: (±2a, a)
Parametric equation: (2at², at²)
Focal length: y + a
When parabola x²= - 4ay
Vertex: (0,0)
Focus: (0, - a)
Axis: x = 0
Directrix: y= a
Length of latus rectum: 4a
Ends of Latus rectum: (±2a, - a)
Parametric equation: (2at², - at²)
Focal length: y - a
When parabola (y - k)²= 4a(x - h)
Vertex: (h+ a, k)
Focus: (0,a)
Axis: y = k
Directrix: x+ a - h= 0
Length of latus rectum: 4a
Ends of Latus rectum: (h+ a, k± 2a)
Parametric equation: (h+ at², k+ 2at)
Focal length: x - h + a
When parabola (x - o)²= 4b(y - q)
Vertex: (p,q)
Focus: (p, b+ q)
Axis: x = p
Directrix: y+ b - q = 0
Length of latus rectum: 4apb
Ends of Latus rectum: (p± 2a, q+ a)
Parametric equation: (p+ 2at, q+ at²)
Focal length: y - q + b
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.
8. Chord joining two points:
The equation 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).
9. LINE AND A PARABOLA :
a) The line y= mx + c meets the parabola y²= 4ax in two points real, coincident or imaginary according as > = < cm => condition of tangency is, c= a/m.
NOTE: Line y= mx + c will be tangent to parabrx²= 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+ m¹)(a - mc)}.
NOTE
Length of the focal chord making an angle α with x-axis is 4a coesec²α.
10. LENGTH OF SUBTANGENT & SUBNORMAL
PT and PG are the tangent and normal respectively at the qP to the parabola y½= 4ax. Then
TN= length of subtangent= 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).
11. 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 & t₂ is [at₁t₂, a(t₁ + t₂)]
12. 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 + at²
NOTE
i) Point of intersection of normals at t₁ & t₂ is [a(t₁²+ t₂²+ t₁t₂+2), - at₁t₂ (t₁ + t₂))
ii) If the normal to the parabola y²= 4ax at the point t₁, meets the parabola again at the point t₂ then t₂ = - (t₁ + 2/t₁).
iii) If normals to the parabola y²= 4ax at the point t₁ & t₂ intersect again on the parabola at the point 't₃' then t₁ t₂ = 2; t₃ = -(r₁ + t₂) and the line joining t₁ and t₂ passes through a fixed point (-2a, 0).
i.e., am³+ m(2a - h)+ k =0
This gives m₁ + m₂+ m₃= 0; m₁ m₂+ m₂m₃+ m₃m₁= (2a - h)/a; m₁m₂m₃= - k/a
Where m₁ , m₂, m₃ are the slopes of the three concurrent normas:
• Algebraic sum of slopes of the three concurrent normals is zero.
• Algebraic sum of ordinates of the three co-normal points on the parabola is zero.
• Centroid of the ∆ formed by three consecutive normal points lies on the axis of parabola (x-axis).
13. 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 functions of x and y then the family of lines will be tangent to the curve Q²= 4PR.
14. PAIR OF TANGENTS:
The equation of the pair of tangents which can be drawn from any point P(x₁ , x₂) 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).
15. DIRECTOR CIRCLE:
Locus of the point of intersection of the perpendicular tangents to the parabola y²= 4ax is called the director circle. It's equation is x+ a = 0 which is parabola's own directrix.
16. 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 us (y₁²- 4ax₁)³⁾²/2a i.e., (S₁)³⁾²/2a, also note that the chord of contact exists only if the point P is not inside.
17. 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≡ 2a(x + x₁) and S₁ = y₁²- 4ax₁.
18 DIAMETER:
The locus of the middle point 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.
19. 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 locus. 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 tangents 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 tangents 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.
f) If the tangent at P and Q meet in T, then:
i) TP and TQ subtend equal angles at the focus S.
ii) ST²= SP. SQ
iii) The triangles SPT and STQ are similar.
g) Tangents and Normals at the Extremities of the latus rectum of a parabola y²= 4ax constitue a square, their points of intersection being (-a,0) and (3a,0).
NOTE
i) The two tangents at the Extremities of focal chord meet on the foot of the directrix.
ii) Figure LNL'G is a square of side 2√2 a.
h) The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.
ELLIPSE
m₂,
≡
