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Question Bank No: 2

1. The lines x cos $α$ + y sin $α$ = p1 and x cos $β$ + y sin $β$ = p2 will be perpendicular if

a)		$α$ = $β$
b)		$\| α - β \|$ │ $\frac{π}{2}$
c)		$α$ = $\frac{π}{2}$
d)		$α$ $\pm$ $β$ = $\frac{π}{2}$

2. The points A ( 0, $\frac{8}{3}$ ), B (1, 3) and C (82, 30) are the vertices of

a)		an acute angled triangle
b)		an isosceles triangle
c)		a right angled triangle
d)		none of these

3. The join of ( -3, 2 ) and (4, 6) is cut by x-axis in the ratio

a)		2: 3 internally
b)		1: 2 externally
c)		1: 3 externally
d)		3:2 internally

4. If a, b, c are in A.P., then the straight line ax + by + c = 0 will always pass through a fixed point whose co-ordinates are

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

5. The equation of the line with slope - $\frac{3}{2}$ and which is concurrent with the lines 4x + 3y -7 = 0 and 8x + 5y - 1 = 0 is

a)		3x + 2y - 63 = 0
b)		3x + 2y - 2 = 0
c)		2y - 3x - 2 = 0
d)		none of these

6. A straight line moves so that the sum of the reciprocals of its intercepts on the co-ordinate axes is unity. Then

a)		the straight line always passes through fixed point (1, 1)
b)		it does not pass through any fixed point
c)		it passes through the origin
d)		none of these

7. Area of the triangle with vertices (a, b), (x1, y1) and (x2, y2) where a, x1, x2 are in G.P. with common ratio r and b, y1, y2 are in G.P. with common ratio s is

a)		ab (r - 1) (s- 1) (s - r)
b)		$\frac{1}{2}$ ab (r + 1) (s + 1) (s- r)
c)		$\frac{1}{2}$ ab (r - 1) (s - 1) (s - r )
d)		ab (r + 1) (s + 1) (r - s)

8. The equations of the sides of a triangle are x = 0, y = m1 x + c1 and y = m2 x + c2. The area of the triangle is

a)		$\frac{C 1 - C 2}{m 1 - m 2}$
b)		$\frac{1}{2 \frac{{(c 1 - c 2)}^{2}}{m 1 - m 2}}$
c)		$\frac{1}{2 \frac{C 1 - C 2}{{(m 1 - m 2)}^{2}}}$
d)		$\frac{{(c 1 - c 2)}^{2}}{{(m 1 - m 2)}^{2}}$

9. A straight line meets the axes at A and B such that the centroid of $Δ$ OAB is (a, a). The equation of the line AB is

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

10. A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y- intercept is

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

11. A point equidistant from the lines 4x + 3y + 10 = 0, 5x - 2y + 26 = 0 and 7x + 24y - 50 = 0 is

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

12. The value of K such that ${3 x}^{2}$ -11xy + ${10 y}^{2}$ - 7x + 13y + K = 0 may represent a pair of straight lines, is

a)		3
b)		4
c)		6
d)		8

13. The equation ${ax}^{2}$ + ${by}^{2}$ + cx + cy = 0, c $\neq$ 0 represents a pair to straight lines if

a)		a + b = 0
b)		b + c = 0
c)		c + a = 0
d)		a + b + c = 0

14. The angle between the lines ${2 x}^{2}$ - 7xy + ${3 y}^{2}$ = 0

a)		$60^{o}$
b)		$45^{o}$
c)		$30^{o}$
d)		$\tan^{- 1} \frac{7}{6}$

15. The pair of straight line ${ax}^{2}$ + 2hxy - ${ay}^{2}$ = 0 and ${bx}^{2}$ = 2gxy - ${by}^{2}$ = 0 such that each bisects the angles between the other, then

a)		ab + gh = 0
b)		$h^{2}$ - ab = 0
c)		ah + bg = 0
d)		ag + bh = 0

16. The equation ${8 x}^{2}$ + 8xy + ${2 y}^{2}$ + 26x + 13y + 15 = 0 represents a pair of parallel straight lines. The distance between them is

a)		$\frac{7}{\sqrt{5}}$
b)		$\frac{7}{\sqrt[2]{5}}$ 7
c)		$\frac{2}{\sqrt{5}}$
d)		none of these

17. The equation ${3 x}^{2}$ + 2hxy + $y^{2}$ = 0 represents a pair of straight lines passing through the origin. The two lines are

a)		real and distinct if $h^{2}$ $>$ 3
b)		real and distinct if $h^{2} >$ 9
c)		real and co-incident if $h^{2}$ = 3
d)		real and co-incident if $h^{2}$ $>$ 3

18. The equation of the circle whose radius is 5 and which touches the circle $x^{2}$ = $y^{2}$ - 2x - 4y - 20 = 0 at the point (5, 5) is

a)		$x^{2}$ + $y^{2}$ + 18x + 16y + 120 = 0
b)		$x^{2}$ + $y^{2}$ - 18x - 16y + 120 = 0
c)		$x^{2}$ + $y^{2}$ - 18 x + 16y + 120 = 0
d)		$x^{2}$ + $y^{2}$ + 18x -16y + 120 = 0

19. If the equation $a x^{2}$ + ${by}^{2}$ + 2hxy + 2gx + 2fy + c = 0 represents a circle, then the condition will be

a)		a = b, c = 0
b)		f = g and h = 0
c)		a = b, h = 0
d)		f = g and c = 0

20. If two circles $x^{2}$ + $y^{2}$ + 2gx+ 2fy = 0 and $x^{2}$ + $y^{2}$ + 2g'x + 2f'y = 0 touch each other, then

a)		gf = g' f'
b)		g' f = gf'
c)		gg' = ff'
d)		none of these

21. The circle $x^{2}$ + $y^{2}$ + 4x - 7y + 12 = 0 cuts an intercept on y-axis equal to

a)		1
b)		3
c)		4
d)		7

22. The equation of the chord of the circle $x^{2}$ + $y^{2}$ - 4x = 0 whose mid-point is (1, 0) is

a)		y = 2
b)		y = 1
c)		x = 2
d)		none of these

23. The value of k for which the circles $x^{2}$ + $y^{2}$ - 3x + ky - y - 9 = 0 becomes concentric is

a)		$\frac{1}{8}$
b)		- $\frac{1}{8}$
c)		$\frac{1}{4}$
d)		- $\frac{1}{4}$

24. Circumcentre of the triangle, whose vertices are (0, 0), (6, 0 ) and (0, 4) is

a)		(2, 0)
b)		(3, 0)
c)		(0, 3)
d)		(3, 2)

25. The length of the chord of the circle $x^{2}$ + $y^{2}$ = 25 joining the points tangents at which intersect at an angle of $120^{o}$ is

a)		$\frac{5}{2}$
b)		5
c)		10
d)		none of these

26. The number of tangents which can be drawn from the point (1, 2) to the circle $x^{2}$ + $y^{2}$ = 5 are

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

27. $x^{2}$ + $y^{2}$ - 6x + 8y - 11 = 0 is a circle. The points (0, 0) and (1, 8) lie,

a)		One on the circle and other outside
b)		Both outside the circle
c)		One outside the circle and one inside
d)		Both inside the circle

28. The equation ${ax}^{2}$ + ${by}^{2}$ + 2hxy + 2gx + 2fy + c = 0 represents a circle only if

a)		a = b, h = 0
b)		a = b $\neq$ 0, h = 0, $g^{2}$ + $f^{2}$ - c $>$ 0
c)		a = b $\neq$ 0, h = 0
d)		a = b $\neq$ 0, h = 0, $g^{2}$ + $f^{2}$ - ac $>$ 0

29. Which of the following lines is a normal to the circle ${(x - 1)}^{2}$ + ${(y - 2)}^{2}$ = 10

a)		x + y = 3
b)		(x - 1) + (y - 2) = 10
c)		x + 2y = 10
d)		2x + y = 3

30. If 2x - 3y = 0 is the equation of the common chord of the circles $x^{2}$ = $y^{2}$ + 4x = 0 and $x^{2}$ + $y^{2}$ + 2 $λ$ y = 0, then the value of $λ$ is

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

31. The number of tangents to the circle $x^{2}$ + $y^{2}$ - 8x - 6y + 9 = 0 which passes through the point (3, -2) is

a)		1
b)		2
c)		0
d)		none of these

32. The point of contact of 3x + 4y + 7 = 0 and $x^{2}$ + $y^{2}$ - 4x - 6y -12 = 0 is

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

33. The radius of the circle inscribed in the triangle by the lines x = 0, y = 0 and 4x + 3y - 24 = 0 is

a)		12
b)		2
c)		2 $\sqrt{2}$
d)		6

34. The line Ax + By + C = 0 will touch the circle $x^{2}$ + $y^{2}$ = $λ$ when

a)		$c^{2}$ = $λ$ ( $A^{2}$ + $B^{2}$ )
b)		$A^{2}$ = $λ$ ( $B^{2}$ + $C^{2}$ )
c)		$B^{2}$ = $λ$ ( $A^{2}$ + $B^{2}$ )
d)		none

35. Let $x^{2}$ + $y^{2}$ = 10x be equation of a circle and let P (7, -11) be a point. P is

a)		on the circle
b)		inside the circle
c)		outside the circle
d)		none

36. Circles $x^{2}$ + $y^{2}$ - 2x- 4y = 0 and $x^{2}$ + $y^{2}$ - 8y - 4 = 0

a)		Touch each other externally
b)		Touch each other internally
c)		Do not touch each other
d)		None of these

37. The length of the tangent from (5, 1) to the circle $x^{2}$ + $y^{2}$ + 6x - 4y - 3 =0 is

a)		81
b)		29
c)		7
d)		21

38. The lines 3x - 4y + 4 + 0 and 6x - 8y - 7 = 0 are tangent to the same circle. The radius of this circle is

a)		$\frac{4}{5}$
b)		$\frac{7}{10}$
c)		$\frac{3}{4}$
d)		$\frac{3}{2}$

39. Four distinct points (2k, 3k), (1, 0), (0, 1) and (0, 0) lie on a circle for

a)		only one value of k
b)		all integral value of k
c)		0 < k < 1
d)		k < 0

40. Given the circles $x^{2}$ + $y^{2}$ - 4x - 5 = 0 and $x^{2}$ + $y^{2}$ + 6x - 2y + 6 = 0. Let P be a point (a, b) such that the tangents from P to both the circles are equal. Then

a)		2a + 10b + 11 = 0
b)		2a - 10b + 11 = 0
c)		10 a - 2b + 11 = 0
d)		10 a + 2b + 11 = 0

41. Given that two circles $x^{2}$ + $y^{2}$ = $r^{2}$ and $x^{2}$ + $y^{2}$ - 10x + 16 = 0, the value of r such that they intersect in real and distinct points is given by

a)		2 < r < 8
b)		r = 2 or r = 8
c)		r < 2 or r < 8
d)		none of these

42. The equation of the circle which touches the axes of co-ordinates and the line x/3 + y/4 = 1 and whose centre lies in the first quadrant is $x^{2}$ + $y^{2}$ - 2cx - 2cy + $c^{2}$ = 0 where c is

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

43. The length of the tangent from (2, 1) to the circle $x^{2}$ + $y^{2}$ + 4y + 3 = 0 is

a)		12
b)		6
c)		$\sqrt{6}$
d)		$\sqrt{12}$

44. A line is drawn through a fixed point P (a, b) to cut the circle $x^{2}$ + $y^{2}$ = $r^{2}$ at A and B, then PA.PB is equal to

a)		${(a + b)}^{2}$
b)		$a^{2}$ + $b^{2}$ - $r^{2}$
c)		${(a - b)}^{2}$ + $r^{2}$
d)		none of these

45. Locus of the point of intersection of lines x cos a + y sin a = a, and x sin a - y cos a = a (a $ϵ$ R) is

a)		$x^{2}$ + $y^{2}$ = $a^{2}$
b)		$x^{2}$ + $y^{2}$ = ${2 a}^{2}$ 2a
c)		$x^{2}$ + $y^{2}$ + 2x + 2y = $a^{2}$
d)		none of these

46. The equation of a line parallel to the tangent to the circle $x^{2}$ + $y^{2}$ = 16 at the point (2, 3) and passing through the origin is

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

47. The number of common tangents to the circles $x^{2}$ + $y^{2}$ + 2x + 8y - 25 = 0 and $x^{2}$ + $y^{2}$ - 4s -10y + 19 = 0, are

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

48. The line 3x - 4y = 0

a)		is a tangent to circle $x^{2}$ + $y^{2}$ = 25
b)		is a normal to the circle $x^{2}$ + $y^{2}$ =25
c)		does not meet the circle $x^{2}$ + $y^{2}$ = 25
d)		does not pass through the origin

49. If the distances from the origin of the centres of the three circles $x^{2}$ + $y^{2}$ - 2 $λ$ i x = $c^{2}$ (i = 1, 2, 3) are in G.P., then the lengths of the tangents drawn to them from any point on the circle $x^{2}$ + $y^{2}$ = $c^{2}$ are in

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

50. If 3x + y = 0 is a tangent to the circle which has its centre at the point (2, 1), then the equation of the other tangent to the circle from the origin

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