The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x² + y² − 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to

(a)

\(0,-\frac { 40 }{ 9 } \)

(b)

0

(c)

\(\frac { 40 }{ 9 } \)

(d)

\(\frac { -40 }{ 9 } \)

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

(a)

\(\frac { 4 }{ 3 } \)

(b)

\(\frac { 4 }{ \sqrt { 3 } } \)

(c)

\(\frac { 2 }{ \sqrt { 3 } } \)

(d)

\(\frac { 3 }{ 2 } \)

The circle x² + y² = 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if

(a)

15< m < 65

(b)

35< m <85

(c)

−85 < m < −35

(d)

−35 < m < 15

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).

(a)

\(\frac { 6 }{ 5 } \)

(b)

\(\frac { 5 }{ 3 } \)

(c)

\(\frac { 10 }{ 3 } \)

(d)

\(\frac { 3 }{ 5 } \)

The radius of the circle 3x² + by² + 4bx − 6by + b² = 0 is

(a)

1

(b)

3

(c)

\( \sqrt {10}\)

(d)

\( \sqrt {11}\)

The centre of the circle inscribed in a square formed by the lines x² − 8x − 12 = 0 and y² − 14y + 45 = 0 is

(a)

(4, 7)

(b)

(7, 4)

(c)

(9, 4)

(d)

(4, 9)

The equation of the normal to the circle x² + y² − 2x − 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is

(a)

x + 2y = 3

(b)

x + 2y + 3 = 0

(c)

2x + 4y + 3 = 0

(d)

x − 2y + 3 = 0

If P(x, y) be any point on 16x² + 25y² = 400 with foci F₁ (3, 0) and F₂ (-3, 0) then PF₁  + PF₂ is

(a)

8

(b)

6

(c)

10

(d)

12

The radius of the circle passing through the point(6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is

(a)

10

(b)

\( {2} \sqrt {5}\)

(c)

6

(d)

4

The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)

(a)

4(a²+b²)

(b)

2(a²+b²)

(c)

a² +b²

(d)

\(\frac { 1 }{ 2 } \)(a²+b²)

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 = r² , then the value of r² is

(a)

2

(b)

3

(c)

1

(d)

4

If x + y = k is a normal to the parabola y² = 12x, then the value of k is

(a)

3

(b)

-1

(c)

1

(d)

9

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) 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

(a)

\(\frac { \sqrt { 2 } }{ 2 } \)

(b)

\(\frac { \sqrt { 3 } }{ 2 } \)

(c)

\(\frac { 1 }{ 2 } \)

(d)

\(\frac { 3 }{ 4 } \)

Tangents are drawn to the hyperbola  \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 4 } =1\) parallel to the straight line 2x − y = 1. One of the points of contact of tangents on the hyperbola is

(a)

\(\left(\frac{9}{2 \sqrt{2}}, \frac{-1}{\sqrt{2}}\right)\)

(b)

\(\left(\frac{-9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)

(c)

\(\left(\frac{9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)

(d)

\((3 \sqrt{3},-2 \sqrt{2})\)

The equation of the circle passing through the foci of the ellipse  \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) having centre at (0, 3) is

(a)

x² + y² − 6y − 7 = 0

(b)

x² + y² − 6y + 7 = 0

(c)

x²+y²−6y−5 = 0

(d)

x²+y²−6y+5 = 0

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 C externally, then the radius of T is equal to

(a)

\(\frac { \sqrt { 3 } }{ \sqrt { 2 } } \)

(b)

\(\frac { \sqrt { 3 } }{ 2 } \)

(c)

\(\frac { 1 }{ 2 } \)

(d)

\(\frac { 1 }{ 4 } \)

Consider an ellipse whose centre is of the origin and its major axis is along x-axis. If its eccentrcity is \(\frac { 3 }{ 5 } \) 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

(a)

8

(b)

32

(c)

80

(d)

40

Area of the greatest rectangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is

(a)

2ab

(b)

ab

(c)

\( \sqrt{ ab}\)

(d)

\(\frac { a }{ b } \)

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

(a)

\(\frac { 1 }{ \sqrt { 2 } } \)

(b)

\(\frac { 1 }{ 2 } \)

(c)

\(\frac { 1 }{ 4 } \)

(d)

\(\frac { 1 }{ \sqrt { 3 } } \)

The eccentricity of the ellipse (x−3)² +(y−4)² \(=\frac { { y }^{ 2 } }{ 9 } \) is

(a)

\(\frac { \sqrt { 3 } }{ 2 } \)

(b)

\(\frac { 1 }{ 3 } \)

(c)

\(\frac { 1 }{ 3\sqrt { 2 } } \)

(d)

\(\frac { 1 }{ \sqrt { 3 } } \)

If the two tangents drawn from a point P to the parabola y² = 4x are at right angles then the locus of P is

(a)

2x + 1 = 0

(b)

x = −1

(c)

2x −1 = 0

(d)

x = 1

The circle passing through (1, -2) and touching the axis of x at (3, 0) passing through the point

(a)

(-5, 2)

(b)

(2, -5)

(c)

(5, -2)

(d)

(-2, 5)

The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is

(a)

a parabola

(b)

a hyperbola

(c)

an ellipse

(d)

a circle

The values of m for which the line y = mx + \(2\sqrt { 5 } \) touches the hyperbola 16x² − 9y² = 144 are the roots of x² − (a + b)x − 4 = 0, then the value of (a+b) is

(a)

2

(b)

4

(c)

0

(d)

-2

If the coordinates at one end of a diameter of the circle x² + y² − 8x − 4y + c = 0 are (11, 2), the coordinates of the other end are

(a)

(-5, 2)

(b)

(-3, 2)

(c)

(5, -2)

(d)

(-2, 5)