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Question Bank No: 3

1. The condition so that the line (x + g) cos $θ$ + (y + f) sin $θ$ = K is a tangent to $x^{2}$ + $y^{2}$ + 2gx + 2fy + c = 0 is

a)		$g^{2}$ + $f^{2}$ = c + $k^{2}$
b)		$g^{2}$ + $f^{2}$ = $c^{2}$ + K
c)		$g^{2}$ + $f^{2}$ = $c^{2}$ + $k^{2}$
d)		$g^{2}$ + $f^{2}$ = c +K

2. The centres of a set of circles, each of radius 3, lie on the circle $x^{2}$ + $y^{2}$ = 25 The locus of any point in the set is

a)		4 $\leq$ $x^{2}$ + $y^{2}$ $\leq$ 64
b)		$x^{2}$ + $y^{2}$ $\leq$ 25
c)		$x^{2}$ + $y^{2}$ $\geq$ 25
d)		3 $\leq$ $x^{2}$ + $y^{2}$ $\leq$ 9

3. If the equation of one tangent to the circle with centre (2, -1) from the origin is 3x + y = 0, then the equation of the other tangent through the origin is,

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

4. A line meets the co-ordinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances of the tangent to the circle at the origin from the points A and B respectively, the diameter of the circle is

a)		m (m +n)
b)		m + n
c)		n (m + n)
d)		$\frac{1}{2}$ (m + n)

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

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

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

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

8. The equation of the directrix of the parabola $x^{2}$ = -4ay is

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

9. The equations x = ${at}^{2}$ , y = 2 at; t $ϵ$ R represent

a)		a circle
b)		an ellipse
c)		a hyperbola
d)		a parabola

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

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

11. The point on the parabola $y^{2}$ = 8x whose distance from the focus is 8, has x co-ordinate as

a)		0
b)		2
c)		4
d)		6

12. The two tangents, perpendicular to each other, to the parabola $y^{2}$ = 4ax intersect on the line

a)		x = a
b)		x + a = 0
c)		x + 2a = 0
d)		x - 2a = 0

13. Three normals are drawn to a parabola $y^{2}$ = 4ax from a given point (x1, y1). The algebraic sum of the ordinates of their feet is

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

14. The parabola $y^{2}$ = 4 ax passes through the point (2, -6), then the length of its latus rectum is

a)		18
b)		9
c)		6
d)		16

15. The line y = 2x + c is a tangent to the parabola $y^{2}$ = 16x if c equals

a)		-2
b)		-1
c)		0
d)		2

16. If the tangent at P and Q on a parabola meet in T, then SP, ST and SQ are in

a)		A.P.
b)		G.P.
c)		H.P
d)		none of these

17. The number of distance normals that can be drawn from ( $\frac{11}{4}$ , 1 $\frac{1}{4}$ ) to the parabola $y^{2}$ = 4x is

a)		3
b)		2
c)		1
d)		4

18. The normals at three points P, Q, R of the parabola $y^{2}$ = 4ax meet in (h, k). The centroid of triangle PQR lies on

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

19. The area of the triangle inscribed in the parabola $y^{2}$ = 4x, the ordinates of whose vertices are 1, 2 and 4 is

a)		7/2 sq units
b)		5/2 sq
c)		/2 sq units
d)		$\frac{3}{4}$ sq units

20. The point on $y^{2}$ = 4ax nearest to the focus has its abcissae equal to

a)		-a
b)		a
c)		a/2
d)		0

21. The slope of a chord of the parabola $y^{2}$ = 4ax which is normal at one end and which subtends a right angle at the origin is

a)		$\pm$ $\frac{1}{\sqrt{2}}$
b)		$\pm$ $\sqrt{2}$
c)		$\pm$ 2
d)		none of these

22. The circle is described on focal radii of a parabola as diameter touches

a)		the axis
b)		the tangent at the vertex
c)		the directrix
d)		none

23. In a parabola semi-latus rectum is the harmonic mean of the

a)		segments of a focal chord
b)		segments of the directrix
c)		segments of a chord
d)		none of these

24. The equation $a x^{2}$ + 2hxy + ${by}^{2}$ + 2gx + 2fy + c = 0 represents an ellipse if

a)		$Δ$ = 0, $h^{2}$ < ab
b)		$Δ$ $\neq$ 0, $h^{2}$ < ab
c)		$Δ$ $\neq$ 0, $h^{2}$ > ab
d)		$Δ$ $\neq$ 0, $h^{2}$ = ab

25. The sum of distances of any point on the ellipse ${3 x}^{2}$ + ${4 y}^{2}$ = 24 from its foci is

a)		8 $\sqrt{2}$
b)		4 $\sqrt{2}$
c)		4 $\sqrt{2}$
d)		none of these

26. If the normal at the one end of a latus-rectum of an ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 passes through one extremity of the minor axis, then the eccentricity of the ellipse is given by the equation

a)		$e^{2}$ + e - 1 = 0
b)		$e^{2}$ + e + 1 = 0
c)		$e^{4}$ + $e^{2}$ + 1 = 0
d)		$e^{4}$ + $e^{2}$ -1 = 0

27. A circle is a limiting case of an ellipse whose eccentricity

a)		tends to a
b)		tends to b
c)		tends to 0
d)		tends to a + b

28. The locus of the point of intersection of perpendicular tangents to the ellipse is called

a)		director circle
b)		auxiliary circle
c)		ellipse itself
d)		similar ellipse

29. Sum of the focal distances of an ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 is equal to

a)		$\frac{1}{\sqrt{2}}$ 1
b)		$\sqrt{\frac{2}{3}}$
c)		$\sqrt{3}$
d)		none of these

30. The number of real tangents that can be drawn to the ellipse ${3 x}^{2}$ + ${5 y}^{2}$ = 32 Passing through (3, 5) is

a)		0
b)		1
c)		2
d)		infinite

31. If tan $θ$ 1 tan $θ$ 2 = $\frac{a^{2}}{b^{2}}$ then the chord joining two points $θ$ 1 and $θ$ 2 on the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 will subtend a right angle at

a)		focus
b)		centre
c)		end of the major axis
d)		end of the minor axis

32. The eccentric angle of a point on the ellipse $\frac{x^{2}}{6}$ + $\frac{y^{2}}{2}$ = 1 whose distances from the centre ellipse is 2, is

a)		$\frac{π}{4}$
b)		$\frac{3 π}{2}$
c)		$\frac{5 π}{3}$
d)		$\frac{7 π}{6}$

33. The locus of the point of intersection of tangents to an ellipse at two points, sum of whose eccentric angles is constant is

a)		a parabola
b)		a circle
c)		an ellipse
d)		a straight line

34. Let E be the ellipse $x^{2}$ + $y^{2}$ = 1 and C be the circle $\frac{x^{2}}{9}$ + $\frac{y^{2}}{4}$ = 9. Let P and Q be the points (1, 2) and (2, 1) respectively, then;

a)		Q lies inside C but outside E
b)		Q lies outside both C and E
c)		P lies inside both C and E
d)		P lies inside C but outside E

35. The eccentricity of the conic ${9 x}^{2}$ - ${16 y}^{2}$ = 144 is

a)		4/5
b)		5/4
c)		4/3
d)		$\sqrt{7}$

36. If e, e' be the eccentricities of two conics S and S' and if $e^{2}$ + ${e'}^{2}$ = 3, then both S and S' can be

a)		ellipses
b)		parabolas
c)		hyperbolas
d)		none of these

37. The equation of the chord joining two points (x1, y1) and (x2, y2) on the rectangular hyperbolas xy = $c^{2}$ is

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