Choose the correct or the most suitable answer from the given four alternatives :
1. The equation of the circle passing through (1, 5) and (4,1) and touching y -axis is x2 + y2 − 5x − 6 y + 9 + λ (4x + 3y −19) = 0 where λ is equal to
(1) 0, - 40/9
(2) 0
(3) 40/9
(4) -40/9
2. The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
(1) 4/3
(2) 4/√3
(3) 2√3
(4) 3/2
3. The circle x2 + y2 = 4x + 8 y + 5 intersects the line 3x − 4 y = m at two distinct points if
(1) 15 < m < 65
(2) 35 < m < 85
(3) -85 < m < -35
(4) -35 < m < 15
4. The length of the diameter of the circle which touches the x -axis at the point (1,0) and passes through the point (2, 3) .
(1) 6/5
(2) 5/3
(3) 10/3
(4) 3/5
5. The radius of the circle 3x2 + by2 + 4bx − 6by + b2 = 0 is
(1) 1
(2) 3
(3) √10
(3) √11
6. The centre of the circle inscribed in a square formed by the lines x2 − 8x −12 = 0 and y2 −14 y + 45 = 0 is
(1) (4, 7)
(2) (7, 4)
(3) (9, 4)
(4) (4, 9)
7. The equation of the normal to the circle x2 + y2 − 2x − 2 y +1 = 0 which is parallel to the line 2x + 4 y = 3 is
(1) x + 2 y = 3
(2) x + 2 y + 3 = 0
(3) 2 x + 4 y + 3 = 0
(4) x - 2 y + 3 = 0
8. If P(x, y) be any point on 16x2 + 25 y2 = 400 with foci F1 (3, 0) and F2 (−3, 0) then PF1 + PF2 is
(1) 8
(2) 6
(3) 10
(4) 12
9. The radius of the circle passing through the point (6, 2) two of whose diameter are x + y = 6 and x + 2 y = 4 is
(1) 10
(2) 2√5
(3) 6
(4) 4
10. The area of quadrilateral formed with foci of the hyperbolas and is
(1) 4(a2 + b2 )
(2) 2(a2 + b2 )
(3) a2 + b2
(4) 1/2 (a2 + b2 )
11. If the normals of the parabola y2 = 4x drawn at the end points of its latus rectum are tangents to the circle (x − 3)2 + ( y + 2)2 = r 2 , then the value of r 2 is
(1) 2
(2) 3
(3) 1
(4) 4
12. If x + y = k is a normal to the parabola y2 = 12x , then the value of k is
(1) 3
(2) -1
(3) 1
(4) 9
13. The ellipse E1 : is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangle R . The eccentricity of the ellipse is
(1) √2/2
(2) √3/2
(3) 1/2
(4) 3/4
14. Tangents are drawn to the hyperbola parallel to the straight line 2x − y = 1. One ofthe points of contact of tangents on the hyperbola is
Ans: (3)
15. The equation of the circle passing through the foci of the ellipse having centre at (0,3) is
(1) x2 + y2 - 6 y - 7 = 0
(2) x2 + y2 - 6 y + 7 = 0
(3) x2 + y2 - 6 y - 5 = 0
(4) x2 + y2 - 6 y + 5 = 0
16. Let C be the circle with centre at (1,1) and radius = 1 . If T is the circle centered at (0, y) passing through the origin and touching the circle Cexternally, then the radius of T is equal to
(1) √3/√2
(2) √3/2
(3) 1/2
(4) 1/4
17. Consider an ellipse whose centre is of the origin and its major axis is along x-axis. If its eccentrcity is 3/5 and the distance between its foci is 6, then the area of the quadrilateral inscribed in the ellipse with diagonals as major and minor axis of the ellipse is
(1) 8
(2) 32
(3) 80
(4) 40
18. Area of the greatest rectangle inscribed in the ellipse is
(1) 2ab
(2) ab
(3) √ab
(4) a/b
19. An ellipse has OB as semi minor axes, F and F ′ its foci and the angle FBF ′ is a right angle. Then the eccentricity of the ellipse is
(1) 1/√2
(2) 1/2
(3) 1/4
(4) 1/√3
20. The eccentricity of the ellipse (x - 3)2 + ( y - 4)2 = y2/9 is
(1) √3 / 2
(2) 1 / 3
(3) 1 / 3√2
(4) 1 / √3
21. If the two tangents drawn from a point P to the parabola y2 = 4x are at right angles then the locus of P is
(1) 2x +1 = 0
(2) x = -1
(3) 2x -1 = 0
(4) x = 1
22. The circle passing through (1, −2) and touching the axis of x at (3, 0) passing through the point
(1) (-5, 2)
(2) (2, -5)
(3) (5, -2)
(4) (-2, 5)
23. The locus of a point whose distance from (−2, 0) is 2/3 times its distance from the line x=-9/2 is
(1) a parabola
(2) a hyperbola
(3) an ellipse
(4) a circle
24. The values of m for which the line y = mx + 2√5 touches the hyperbola 16x2 − 9 y2 = 144 are the roots of x2 − (a + b)x − 4 = 0 , then the value of (a + b) is
(1) 2
(2) 4
(3) 0
(4) -2
25. If the coordinates at one end of a diameter of the circle x2 + y2 − 8x − 4 y + c = 0 are (11, 2) , the coordinates of the other end are
(1) (-5, 2)
(2) (2, -5)
(3) (5, -2)
(4) (-2, 5)
Choose the correct or the most suitable answer from the given four alternatives :
1. The equation of the circle passing through (1, 5) and (4,1) and touching y -axis is x2 + y2 − 5x − 6 y + 9 + λ (4x + 3y −19) = 0 where λ is equal to
(1) 0, - 40/9
(2) 0
(3) 40/9
(4) -40/9
2. The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
(1) 4/3
(2) 4/√3
(3) 2√3
(4) 3/2
3. The circle x2 + y2 = 4x + 8 y + 5 intersects the line 3x − 4 y = m at two distinct points if
(1) 15 < m < 65
(2) 35 < m < 85
(3) -85 < m < -35
(4) -35 < m < 15
4. The length of the diameter of the circle which touches the x -axis at the point (1,0) and passes through the point (2, 3) .
(1) 6/5
(2) 5/3
(3) 10/3
(4) 3/5
5. The radius of the circle 3x2 + by2 + 4bx − 6by + b2 = 0 is
(1) 1
(2) 3
(3) √10
(3) √11
6. The centre of the circle inscribed in a square formed by the lines x2 − 8x −12 = 0 and y2 −14 y + 45 = 0 is
(1) (4, 7)
(2) (7, 4)
(3) (9, 4)
(4) (4, 9)
7. The equation of the normal to the circle x2 + y2 − 2x − 2 y +1 = 0 which is parallel to the line 2x + 4 y = 3 is
(1) x + 2 y = 3
(2) x + 2 y + 3 = 0
(3) 2 x + 4 y + 3 = 0
(4) x - 2 y + 3 = 0
8. If P(x, y) be any point on 16x2 + 25 y2 = 400 with foci F1 (3, 0) and F2 (−3, 0) then PF1 + PF2 is
(1) 8
(2) 6
(3) 10
(4) 12
9. The radius of the circle passing through the point (6, 2) two of whose diameter are x + y = 6 and x + 2 y = 4 is
(1) 10
(2) 2√5
(3) 6
(4) 4
10. The area of quadrilateral formed with foci of the hyperbolas and is
(1) 4(a2 + b2 )
(2) 2(a2 + b2 )
(3) a2 + b2
(4) 1/2 (a2 + b2 )
11. If the normals of the parabola y2 = 4x drawn at the end points of its latus rectum are tangents to the circle (x − 3)2 + ( y + 2)2 = r 2 , then the value of r 2 is
(1) 2
(2) 3
(3) 1
(4) 4
12. If x + y = k is a normal to the parabola y2 = 12x , then the value of k is
(1) 3
(2) -1
(3) 1
(4) 9
13. The ellipse E1 : is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangle R . The eccentricity of the ellipse is
(1) √2/2
(2) √3/2
(3) 1/2
(4) 3/4
14. Tangents are drawn to the hyperbola parallel to the straight line 2x − y = 1. One ofthe points of contact of tangents on the hyperbola is
Ans: (3)
15. The equation of the circle passing through the foci of the ellipse having centre at (0,3) is
(1) x2 + y2 - 6 y - 7 = 0
(2) x2 + y2 - 6 y + 7 = 0
(3) x2 + y2 - 6 y - 5 = 0
(4) x2 + y2 - 6 y + 5 = 0
16. Let C be the circle with centre at (1,1) and radius = 1 . If T is the circle centered at (0, y) passing through the origin and touching the circle Cexternally, then the radius of T is equal to
(1) √3/√2
(2) √3/2
(3) 1/2
(4) 1/4
17. Consider an ellipse whose centre is of the origin and its major axis is along x-axis. If its eccentrcity is 3/5 and the distance between its foci is 6, then the area of the quadrilateral inscribed in the ellipse with diagonals as major and minor axis of the ellipse is
(1) 8
(2) 32
(3) 80
(4) 40
18. Area of the greatest rectangle inscribed in the ellipse is
(1) 2ab
(2) ab
(3) √ab
(4) a/b
19. An ellipse has OB as semi minor axes, F and F ′ its foci and the angle FBF ′ is a right angle. Then the eccentricity of the ellipse is
(1) 1/√2
(2) 1/2
(3) 1/4
(4) 1/√3
20. The eccentricity of the ellipse (x - 3)2 + ( y - 4)2 = y2/9 is
(1) √3 / 2
(2) 1 / 3
(3) 1 / 3√2
(4) 1 / √3
21. If the two tangents drawn from a point P to the parabola y2 = 4x are at right angles then the locus of P is
(1) 2x +1 = 0
(2) x = -1
(3) 2x -1 = 0
(4) x = 1
22. The circle passing through (1, −2) and touching the axis of x at (3, 0) passing through the point
(1) (-5, 2)
(2) (2, -5)
(3) (5, -2)
(4) (-2, 5)
23. The locus of a point whose distance from (−2, 0) is 2/3 times its distance from the line x=-9/2 is
(1) a parabola
(2) a hyperbola
(3) an ellipse
(4) a circle
24. The values of m for which the line y = mx + 2√5 touches the hyperbola 16x2 − 9 y2 = 144 are the roots of x2 − (a + b)x − 4 = 0 , then the value of (a + b) is
(1) 2
(2) 4
(3) 0
(4) -2
25. If the coordinates at one end of a diameter of the circle x2 + y2 − 8x − 4 y + c = 0 are (11, 2) , the coordinates of the other end are
(1) (-5, 2)
(2) (2, -5)
(3) (5, -2)
(4) (-2, 5)
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