EXERCISE - A
a) 3y²= 5x. (0,0),(5/12,0),5/3,y= 0, 12x+5= 0
b) 2x² + 3y= 0. (0,0),(0,-3/8),3/2,x=0, 8y -3= 0
c) (y -3)²= 6(x -2). (2,3),(7/2,3), 6,y=3, x= 1/2
d) (x-3)²+ 8(y+1)=0. (3,1),(3,-3), 8,x=3, y= 1
e) y²+12 = 4x+ 4y. (2,2),(3,2), 4,y=2, x= 1
f) 4y² - 20x - 8y + 39= 0. (7/4,1),(3,1), 5,y=2, x= 1/2
g) y= x²- 3x +4. (3/2,7/4),(3/2,2), 1, 2x-3= 0, 2y-3= 0
h) 2y²- 4y +5 = x. (3,1),(25/8,2), 1/2 ,y=1, 8x -23= 0
2) a) Find the point on the parabola y²= 12x at which the ordinate is double the abscissa. (3,6)
b) Find the point on the parabola y²= -36 x at which the ordinate is three times the abscissa. (-4,-12)
3) Find the coordinates of focus and the length of latus rectum of the parabola y²= 2mx which passes through the point of intersection of the straight line x/a + y/b =1 and x/b + y/a =1.
4) The parabola x²+ 2py =0 passes through the point (4,-2); find the coordinates of focus and the length of latus rectum. (0,-2); 8
5) Find the coordinates of focus and the length of latus rectum of the parabola y²= 2ax which passes through the point of intersection of the straight line x/3 + y/2 =1 and x/2 + y/3 =1.
6) Find the coordinates of focus and the length of latus rectum of the parabola y²= 4ax which passes through the point of intersection of the straight line 3x + y =-5 and x + 3y =1.
7) The parabola y²= 4ax which passes through the centre of the circle 2x²+ 2y²- 4x +12y-1 =0. Find the coordinates of the focus , length of the latus rectum. (9/4,0)9, 4x+9=0
8) The parabola y²= 4ax which passes through the centre of the circle x²+ y²+4x -12y-4 =0. Find the length of the latus rectum. 18
9) Find the focal distance of a point on the parabola x²= 8y if the ordinates of the point be 11. 13
10) Find a point on the parabola y²= 8x whose focal distance is 4. If the focal distance of a point on the parabola be 8, then find the co-ordinate of that point. (2,±4),(6,±4√3)
11) The focal distance of a point on the parabola y²=12x is 6; find the coordinates of the point. (3,6) and (3,-6)
12) find the point on the parabola y²= 16x whose distance from the dielectrix is 6 units. (2,±4√2)
13) ) find the equation of the parabola passing through (8,2) whose vertex is at origin and axis is y-axis. Find the distance of that point from the focus of the parabola. x²= 32y
14) the parabola x²= 4ay passes through the centre of the circle x²+ y²-8x +4y -3= 0. Find the coordinates of the focus and the length of the latus rectum of the parabola. (0,-2),8 units
15) The parabola passes through y²= 2ax passes through the centre of the circle 4x²+ 4y² - 8x +12y-7=0. Find the focus, length of the latus rectum and the equation of the directrix of the parabola. (9/16,0),9/4, 16x+9= 0
16) The parabola y²= 4ax passes through the point of intersection of the line 3x+ y+5= 0 and x+ 3y-1= 0. Find the coordinates of the focus and the length of the latus rectum of the parabola. (-1/8,0),1/2
17) A parabola passes through the point (3,2) and (-2,-1) and its axis is the x-axis. Find the equation of the parabola. 5y²= 3x +11
EXERCISE - B
1) Find the equation of the parabola whose coordinates of vertex the focus are (-2, 3) and (1, 3) respectively. y²- 6y - 12x =15.
2) Find the equation of the parabola whose vertex is at (1,2) and focus is at (-1,2).
(y-2)²+8(x- 1) = 0
3) The coordinates of the vertex and focus are respectively (2,3) and (2,-1). Show that the question of the parabola is x²- 4x +16y =44.
4) Find the equation of the parabola whose co-ordinates of vertex and focus (0,0) and (3/2,0) respectively. y²= 6x
5) the vertex of a parabola is at the origin and its focus is (0, -5/4); find the equation of the parabola. x²+ 5y=0
6) A parabola having vertex at the origin and axis along x-axis passes through (6,-2); find the equation of the parabola. 3y²= 2x
7) The axis of a parabola is along y-axis and vertex is (0,0). If it passes through (-3,2). Find the co-ordinates of its focus. x²+ 16y=0.
8) Find the equation of the parabola whose vertex is at (-2,3) axis is parallel to x-axis and length of lactus rectum is 12. (y +3)²= 12(x -2)
9) The coordinates of the vertex and focus of a parabola are (1,2) and (-1,2) respectively; find its equation. (y -2)²+ 8(x +1)=0
10) Show that the equation of the parabola whose vertex is (2,3) and focus is (2,-1) is x²- 4x + 16y = 44.
EXERCISE - C
1) Determine the positions of the point
a) (3,6) b) (4,3) c) (1,-3) with respect to the parabola y²= 9x. Outside, Inside, on
2) Examine with reasons the validity of the following statement: "The point (4,3) lies outside the parabola y²= 4x but the point (-4,-3) lies within it".
3) For what values of a will the point (8,4) be an inside point of the parabola y²= 4ax ? a> 1/2
EXERCISE - D
1) Find the equation of the parabola whose vertex is at (4,-2), length of the latus rectum is 8 and axis is y+ 2=0. (y+2)²= ± 8(x -4)
2) Find the equation of the parabola whose vertex is at (-2,3) and the equation of the directrix is x +7=0. (y-3)²= 20(x +2)
3) Find the equation of the parabola was vertex is (0,0) and directrix is the line x + 3 = 0. y²= 12x
4) Find the equation of the parabola whose vertex is at the origin and directrix is the line y - 4 =0. x²+ 16y=0
5) Find the equation of the parabola whose focus is at the origin and the equation of the directrix is x + y=1. x²- 2xy + y²+ 2x + 2y -1=0
6) Find the equation of the parabola and the coordinates of its focus if the vertex of the parabola is at (-1,1) and the directrix is x + y + 4=0. (0,0), x²- 2xy + y² -8x -8y -16 =0
7) Find the vertex, length of the latus rectum and the axis of the parabola whose focus is at the point (3,4) and directrix is 3x + 4y +25=0. (0,0),20 units, 4x = 3y
8) Find the equation of the parabola whose focus is (2,1) and whose directrix is 3x - y+ 1 = 0. x²+ 9y²+ 6xy - 46x -18y +49=0.
9) The equation of the directrix of a parabola x= y and the coordinates of its focus are (4,0). Find the equation of the parabola. x²+ y²+ 2xy - 16x + 32 =0.
10) Find the co-ordinates of vertex and the length of latus of the parabola whose focus is (0,0) and the directrix is the line 2x + y=1. (1/5,1/10) and 2/√5
*11) Find the equation of the parabola whose coordinates of vertex are (-2,3) and the equation of the directrix is 2x + 3y+8=0.
*12) t6he directrix of the parabola is x + y + 4 = 0 and vertex is the point (-1,-1). Find
a) the position of focus. (0,0)
b) the equation of the parabola. x²+ y² - 2xy - 8x -8y -16 =0.
EXERCISE - E
1) Find the equation of the parabola whose vertex is (-1,3) and focus is (3,-1). 2{(x -3)²+ (y +1)²}= (x - y +12)²
2)
) Find the equation of the circle whose diameter is the line segment joining the focus of the parabola y²= 12x and the centre of the circle x²+ y²-18x- 16y+45= 0. x²+ y²-12x- 8y+27=0
) Find the equation of the circle whose diameter is the latus rectum of the parabola y²-4ax = 0 and prove that the circle passes through the point of intersection of the axis and the directrix of the parabola. x²+ y²-2ax- 3a²=0
) Find the equation of the circle passing through the origin and through the foci of the two parabolas y²= 8x and x²= 24y. x²+ y²-2x- 6y=0
) Find two points in the parabola x² = 8y, each of which is at a distance of 4 units from the focus. Find also the equation of the circle whose diameter is the line segment joining the two points. (4,2),(-4,2) , x²+ y²- 4y-12=0
) A circle is drawn through the vertex and the two ends of the latus rectum of a parabola. If the length of the latus rectum be 2m, find the radius of the circle. 5m/4
) if the vertex and the focus of a parabola are on the x-axis and at distance of a and b respectively from the origin, then show that the equation of the parabola is y²= 4(b - a)(x - a).
) The vertex of a parabola is at (-4,2), axis is parallel to y-axis and length of the latus rectum is 8; find the equation of the parabola. (x+4)²= 8(y -2) or (x +4)²+8(y -2)=0
) A parabola has its axis parallel to x-axis and passes through the points (2,0),(1,-1) and (-2,-6). Find the equation of the parabola. 15x= y²+16y +30
) Find the equation of the parabola passing through the point (3,0),(- 3,0)( 2,5) and having its axis parallel to the y-axis. Also find the coordinate of its vertex. x²+ y =9,(0,9)
) The equation of the directrix of a parabola is x= y and the co-ordinates of its focus are (4,0); find the equation of the parabola. x²+ 2xy + y²-16x + 32=0
) Find the equation of the parabola whose focus is at (5,3) and vertex is at (3,1). x²- 2xy + y²- 20x - 12y +68 =0
) The coordinates of the focus and the vertex of a parabola are respectively (6,2) and (4,3). Find the equation of the parabola. x²+ 4xy + 4y²- 60x - 20y +200 =0
) The vertex of a parabola is at (2,-3) and the equation of the latus rectum is x - y +5=0; find the equation of the parabola. x²+ 2xy + y²+ 42x -38y -199 =0
) if the coordinates of one end of the latus rectum of a parabola are (4,-1) and the co-ordinate of the point of intersection of the axis in the lactus rectum are (4,3), find the equation of the parabola. y² -8x -6y+25 =0 or y²+ 8x -6y -39 =0
) Prove that, if the coordinates of the point are x= 1 - 2t, y= 3t²- 2 (t is parameter), the locus of the point is a parabola. Find the vertex and the length of the latus rectum of the parabola. (1,-2), 4/3 units
) if k be a variable parameter, show that the locus point x = 2 sec k - 1, y = -2 tan² k is a parabola. Find the vertex and the length of the latus rectum of the parabola. M(-1,2), 2 units
) A double ordinate of the parabola y²= 4ax subtends a right angle at the vertex. Find the length of that double ordinate. 8a
) If the length of a double ordinate of the parabola 2y²= 3x is 6, find the equation of the circle whose diameter is that double ordinate. x² + y²- 12x + 27 =0
) Find the equation of the chord of the parabola y²= 4ax passing through the points (at₁², 2at₁) and (at₂², 2at₂). 2x - y(t₁ + t₂)+ 2at₁t₂ =0.
) if the lines joining the vertex with the two points (at₁², 2at₁) and (at₂², 2at₂) on the parabola y²= 4ax are at right angle, prove that t₁t₂ + 4=0.
) Show that the equation of the chord of the parabola x²= 4ay through the point (x₁, y₁) and (x₂, y₂) on it, is (x - x₁)(x - x₂)=nx²- 4ay.
) Prove that if the co-ordinates of one end of the focal chord of the parabola y²= 4ax (at², 2at) then the co-ordinates of other end of the chord will be (a/t², - 2a/t).
) A straight line through the focus of the parabola y²= 4ax intersect the parabola at the point (x₁², y₁) and (x₂, y₂) show that x₁ x₂ =a².
) A focal chord SE of the parabola y²= 8x passes through the point end point, having positive coordinates, of another EE' : x =4. Find the equation and the length of the chord. y= 2√2 (x -2), 9 units
) Q is any point on y²= 4ax, ordinate of the point Q is QN, P is the midpoint of QN; prove that the locus of the point P is a parabola, the length of whose latus rectum is one-fourth of the length of the latus rectum.
) PN is any ordinate of the parabola y²= 4ax; the point M of divides PN in the ratio m: n. Find the locus of M. (m+ n)²y²= 4an²x
) Show that the locus of the middle point of chords of the parabola y²= 4ax which pass through the vertex is the parabola y²= 2ax.
) If P(3, 3/2) is a fixed point and Q is a variable point on the parabola x²= 6y, find the locus of the point which divides PQ in the ratio 1:2. x²- 4x - 2y +6=0
) A parabola arch span 60cms and a height of 45 cms. Find the distance from the central axis the point on the arch whose height is 25cms. 20 cms
) Prove that the circle drawn on the focal cord of a parabola as diameter touches its directrix .
) The equation of the axis and the directrix of a parabola are y - 3=0 and x +3=0 respectively and the length of the latus rectum is 8 units. Find the equation and the vertex of the parabola. (y -3)²= 8(x +1),(-1,3) or (y -3)²+ 8(x +5)= 0, ,(-5,3)
) Find the equation of the parabola whose focus is at (0,-4), axis is parallel to x-axis and the length of the latus rectum is 16 units. (y +4)²= 16(x +4), or (y +4)²+16(x -4)=0
) Find the locus of the middle points of those chords of the parabola y²= 4ax which pass through a fixed point (m,n). y²- ny = 2a(x - m)
) Find the locus of the middle points of those chords of the parabola y²= 4ax which subtends right angles at its vertex . y²= 2a(x - 4a)
) P, Q, R are three points on the parabola y²= 4ax such that PQ passes through the focus and PR is perpendicular the axis . Show that the locus of the midpoint of QR is y²= 2a(x +a)
Objective and Short Answer Type Questions
1) If the co-ordinates of the focus and the vertex of a parabola are (4,3)!and (1,-1) respectively, find the point of intersection of axis and directrix and the length of the latus rectum. (-2,-5), 20 units
2) if the point of intersection of the axis and the directrix of a parabola and the vertex are (6,5) and (4,1) respectively, find the distance of the point (-1,-2) from the focus. √10 units
3) Find such a point on the parabola y²= -9x that the ordinate of the point is 3 times of its abscissa . (-1,-3)
4) If the parabola x²+ 5py=0 passes through the point (3,-3), find the coordinates of the focus and the length of the latus rectum. (0,-3/4)
5) What is the distance of the point (-2,6) on the parabola y² + 18x = 0 from the focus ? 13/2 units
6) Find the equation of the axis of the parabola whose focus is at (3,4) and the equation of the directrix is 3x + 4y+25=0. 4x - 3y=0
7) Find the equation of the parabola whose vertex is at origin, axis is x-axis and which passes through the point (-3,4). 3y²+ 16x =0
8) Find the equation of the parabola whose vertex is at the origin, axis is y-axis and which passes through the point (-3,4). 4x²= 9y
9) Find the equation of the parabola whose focus is at (6,0) and directrix is x= 0. y²= 12(x -3)
10) If the vertex of a parabola is at the origin and its directrix is 2x+ 5=0, find its equation . y²= 10x
11) If the vertex of a parabola is at the origin and its directrix is 3y -7=0, find the equation of the parabola. 3x²+ 28y=0
12) Find the length of the latus rectum of the parabola 5x²+ 30x + 2y +59=0. 2/5
13) Find the length of the latus rectum of the parabola y = - 2x²+ 12x -17. 1/2
14) Show that the latus rectum of a parabola is the third proportional of the abscissa and ordinate of a point on the parabola.
15) A straight line parallel to the axis of a parabola interesects the parabola to one point /two points/more than two points. Which one is correct ?
16) what type of conic section is the locus of the moving point (at², 2at)? Find the equation of the locus. Parabola, y²= 4ax.
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