Untitled Document

Question Bank No: 1

1. The equation of the parabola with directrix x=2 and the axis y=0 is

a)		$y^{2} = 8 x$
b)		$y^{2} = - 8 x$
c)		$y^{2} = 4 x$
d)		$y^{2} = - 4 x$

2. The equation of the parabola with focus at (0,3) and the directrix y+3=0 is

a)		$y^{2} = 12 x$
b)		$y^{2} = - 12 x$
c)		$x^{2} = 12 y$
d)		$y^{2} = - 4 x$

3. Equation of the directrix of the parabola $x^{2} = - 4 ay$ is

a)		x+q=0
b)		x-a=0
c)		y+a=0
d)		y-a=0

4. The latus rectum of the parabola $x^{2} - 4 x - 2 y - 8 = 0$ is

a)		8
b)		4
c)		2
d)		1

5. The eccentricity of the parabola $y^{2} = - 8 x$ is

a)		-2
b)		2
c)		-1
d)		1

6. The vertex of the parabola $y^{2} = 4 (x + 1) is$

a)		(0,1)
b)		(0,-1)
c)		(1,0)
d)		(-1,0)

7. The locus of the points which are equidistance from (-9,0) and x=a is

a)		$y^{2} = 4 ax$
b)		$y^{2} = - 4 ax$
c)		$x^{2} + 4 ay = 0$
d)		$x^{2} - 4 ay = 0$

8. The length of the latus rectum of the parabola 4 $y^{2} + 2 x - 20 y + 17 = 0$ is

a)		3
b)		6
c)		1/2
d)		0

9. The equation of the directix of the parabola 5 $y^{2} = 4 x$ is

a)		4x-1=0
b)		4x+1=0
c)		5x+1=0
d)		5x-1=0

10. The axis of the parabola $x^{2} - 3 y - 6 x + 6 = 0$ is

a)		x= -3
b)		y=-1
c)		x=3
d)		y=1

11. A particle projected from the ground just clears a wall of height b at a distance a and strikes the ground at a distance c from the point of projection. The angle of projection is

a)		$45^{\circ}$
b)		$\tan^{- 1} \frac{bc}{a (c - a)}$
c)		$\tan^{- 1} \frac{bc}{a}$
d)		$\tan^{- 1} \frac{bc}{ca}$

12. The equation of a tangent to the parabola $y^{2} = 8 x$ is $y = x + 2.$ The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

a)		(0, 2)
b)		(2, 4)
c)		(-2, 0)
d)		(-1, 1)

13. ABCD is a square with side AB=2. A point P moves such that its distance from A equals its distance from the line BD. The locus of P meets the line AC at $T_{1}$ and the line through A parallel to BD at $T_{2}$ and $T_{3}$ . The area of $Δ {{T_{1} T}_{2} T}_{3}$ is

a)		$\frac{1}{2}$
b)		$\frac{2}{3}$
c)		1
d)		2

14. The equation of the common tangents to the parabola y= $x^{2}$ and $y = - {(x - 2)}^{2}$ are

a)		$y = 4 (x - 1)$
b)		$y = 4$
c)		$y = - 4 (x - 1)$
d)		$y = - 30 x - 50$

15. The axis of a parabola is along the line y=x and the distance of its vertex from the origin is $\sqrt{2}$ and that of its focus from the origin is 2 $\sqrt{2} .$ If the vertex and focus lie in the frist quadrant, the equation of the parabola is

a)		${(x + y)}^{2} = x - y - 2$
b)		${(x - y)}^{2} = x + y - 2$
c)		${(x - y)}^{2} = 4 (x + y - 2)$
d)		${(x - y)}^{2} = 8 (x + y - 2)$

16. The parabola y= $x^{2} - 8 x + 15$ cuts the x-axis at P and Q. A circle is drawn through P and Q so that the origin is outside it. The length of a tangent to the circle from O is

a)		15
b)		$\sqrt{8}$
c)		$\sqrt{15}$
d)		8

17. If two distinct chords of a parabola $y^{2} = 4 ax$ passing through the point (a, 2a) are bisected by the line x+y=1, then the length of the latus-rectum can be

a)		2
b)		7
c)		4
d)		5

18. If the normals at the end points of a variable chord PQ of the parabola $y^{2} - 4 y - 2 x = 0$ are perpendicular, then the tangents at P and Q will intersect on the line

a)		$x + y = 3$
b)		$3 x - 7 = 0$
c)		$y + 3 = 0$
d)		$2 x + 5 = 0$

19. If $a \neq 0$ and the line $2 bx + 3 cy + 4 d = 0$ passes through the points of intersection of the parabolas $y^{2} = 4 ax$ and $x^{2} = 4 ay,$ then

a)		$d^{2} + {(2 b + 3 c)}^{2} = 0$
b)		$d^{2} + {(3 b + 2 c)}^{2} = 0$
c)		$d^{2} + {(2 b - 3 c)}^{2} = 0$
d)		$d^{2} + {(3 b - 2 c)}^{2} = 0$

20. The normal at the point $(b {t_{2}}^{2}, 2 b t_{2}),$ then

a)		$t_{2} = - t_{1} - \frac{2}{t_{1}}$
b)		$t_{2} = - t_{1} + \frac{2}{t_{1}}$
c)		$t_{2} = t_{1} - \frac{2}{t_{1}}$
d)		$t_{2} = t_{1} + \frac{2}{t_{1}}$

21. Common tangents to the circle $x^{2} + y^{2} = 2 a^{2} and$ parabola $y^{2} = 8 ax$ are

a)		$x = \pm (y + 2 a)$
b)		$y = \pm (x + 2 a)$
c)		$x = \pm (y + a)$
d)		$y = \pm (x + a)$

22. The angle between the tangents drawn from the point (1, 4) to the parabola $y^{2} = 4 x$ is

a)		$\frac{π}{6}$
b)		$\frac{π}{4}$
c)		$\frac{π}{3}$
d)		$\frac{π}{2}$

23. The curve described parametrically by $x = t^{2} + t + 1, y = t^{2} - t + 1$ represents

a)		a pair of lines
b)		an ellipse
c)		a parabola
d)		a hyperbola

24. The equation of the common tangent touching the circle ${(x - 3)}^{2} + y^{2} = 9$ and the parabola $x^{2} = 4 x$ above the x-axis is

a)		$\sqrt{3} y = 3 x + 1$
b)		$\sqrt{3} y = - (x + 3)$
c)		$\sqrt{3} y = x + 3$
d)		$\sqrt{3} y = - (3 x + 1)$

25. The equation of the tangent to the curves $y^{2} = 8 x$ and $xy = - 1$ is

a)		$3 y = 9 x + 2$
b)		$y = 2 x + 1$
c)		2 $y = x + 8$
d)		$y = x + 2$

26. The slopes of tangents to the circle ${(x - 6)}^{2} + y^{2} = 2$ which passes through the focus of the parabola $y^{2} = 16 x$ are

a)		$\pm 2$
b)		$\frac{1}{2}, - 2$
c)		$- \frac{1}{2}, 2$
d)		$\pm 1$

27. The locus of the midpoint of the line segment joining the focus to a moving point on the parabola $y^{2} = 4 ax$ is another parabola with directrix

a)		$x = - a$
b)		$x = - \frac{a}{2}$
c)		$x = 0$
d)		$x = \frac{a}{2}$

28. If $x + y = k$ is normal to $y^{2} = 12 x,$ then k is

a)		3
b)		9
c)		4
d)		$\frac{1}{4}$

29. The equation of the directrix of the parabola $y^{2} + 4 y + 4 x + 2 = 0$ is

a)		$x = - 1$
b)		$x = 1$
c)		$x = - \frac{3}{2}$
d)		$x = \frac{3}{2}$

30. The centre of the circle passing through the point (0, 1) and touching the curve $y = x^{2}$ at (2, 4) is

a)		$(\frac{- 16}{5}, \frac{27}{10})$
b)		$(\frac{- 16}{7}, \frac{53}{10})$
c)		$(\frac{- 16}{5}, \frac{53}{10})$
d)		none of these