1. The equation \(a x^2+2 h x y+b y^2+2 g x+2 f y+c=0\) represents a rectangular hyperbola if

a) \(\Delta \neq 0, h^2>a b, a+b=0\)

b) \(\Delta \neq 0, h^2<a b, a+b=0\)

c) \(\Delta \neq 0, h^2=a b, a+b=0\)

d) None of these

2. The line passing through the extremity \(A\) of the major axis and extremity \(B\) of the minor axis of the ellipse \(x^2+9 y^2=9\) meets its auxiliary circle at the point \(M\). Then, the area of the triangle with vertices at \(A, M\) and the origin \(O\) is

a) \(\frac{31}{10}\)

b) \(\frac{29}{10}\)

c) \(\frac{21}{10}\)

d) \(\frac{27}{10}\)

3. From the point \((-1,-6)\) two tangents are drawn to the parabola \(y^2=4 x\). Then, the angle between the two tangents is

a) \(30^{\circ}\)

b) \(45^{\circ}\)

c) \(60^{\circ}\)

d) \(90^{\circ}\)

4. The centre of the ellipse \(4 x^2+9 y^2+16 x-18 y-11=0\) is

a) \((-2,-1)\)

b) \((-2,1)\)

c) \((2,-1)\)

d) None of these

5. The circle whose equation are \(x^2+y^2+c^2=2 a x\) and \(x^2+y^2+c^2-2 b y=0\) will touch one another externally if

a) \(\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{a^2}\)

b) \(\frac{1}{c^2}+\frac{1}{a^2}=\frac{1}{b^2}\)

c) \(\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}\)

d) None of these

6. In an ellipse the distance between the foci is 8 and the distance between the directrices is 25 . The length of major axis is

a) \(10 \sqrt{2}\)

b) \(20 \sqrt{2}\)

c) \(30 \sqrt{2}\)

d) None of these

7. If \(l x+m y+n=0\) represents a chord of the ellipse \(b^2 x^2+a^2 y^2=a^2 b^2\) whose eccentric angles differ by \(90^{\circ}\), then

a) \(a^2 l^2+b^2 m^2=n^2\)

b) \(\frac{a^2}{l^2}+\frac{b^2}{m^2}=\frac{\left(a^2-b^2\right)^2}{n^2}\)

c) \(a^2 l^2+b^2 m^2=2 n^2\)

d) None of these

8. If the latusrectum of a hyperbola forms an equilateral triangle with the vertex at the centre of the hyperbola, then the eccentricity of the hyperbola is

a) \(\frac{\sqrt{5}+1}{2}\)

b) \(\frac{\sqrt{11}+1}{2}\)

c) \(\frac{\sqrt{13}+1}{2 \sqrt{3}}\)

d) \(\frac{\sqrt{13}-1}{2 \sqrt{3}}\)

9. The eccentricity of the conic \(4 x^2+16 y^2-24 x-32 y=1\) is

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

b) \(\sqrt{3}\)

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

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

10. If the chords of contact of tangents from two points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\) to the hyperbola \(4 x^2\) \(-9 y^2-36=0\) are at right angles, then \(\frac{x_1 x_2}{y_1 y_2}\) is equal to

a) \(\frac{9}{4}\)

b) \(-\frac{9}{4}\)

c) \(\frac{81}{16}\)

d) \(-\frac{81}{16}\)

11. The equation of a circle which cuts the three circles

\[

\begin{aligned}

& x^2+y^2-2 x-6 y+14=0 \\

& x^2+y^2-x-4 y+8=0 \\

& x^2+y^2+2 x-6 y+9=0

\end{aligned}

\]

orthogonally, is

a) \(x^2+y^2-2 x-4 y+1=0\)

b) \(x^2+y^2+2 x+4 y+1=0\)

c) \(x^2+y^2-2 x+4 y+1=0\)

d) \(x^2+y^2-2 x-4 y-1=0\)

12. The length of the common chord of the ellipse \(\frac{(x-1)^2}{9}+\frac{(y-2)^2}{4}=1\) and the circle \((x-1)^2+\) \((y-2)^2=1\) is

a) 2

b) \(\sqrt{3}\)

c) 4

d) None of these

13. The mirror image of the directrix of the parabola \(y^2=4(x+1)\) in the line mirror \(x+2 y=3\), is

a) \(x=-2\)

b) \(4 y-3 x=16\)

c) \(x-3 y=0\)

d) \(x+y=0\)

14. The line \(x=a t^2\) meets the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) in the real points, if

a) \(|t|<2\)

b) \(|t| \leq 1\)

c) \(|t|>1\)

d) None of these

15. The length of the latusrectum of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1\), is

a) \(\frac{2 a^2}{b}\)

b) \(\frac{2 b^2}{a}\)

c) \(\frac{b^2}{a}\)

d) \(\frac{a^2}{b}\)

16. The condition that the chord \(x \cos \alpha=0+y \sin \alpha-p=0\) of \(x^2+y^2-a^2=0\) may subtend a right angle at the centre of the circle is

a) \(a^2=2 p^2\)

b) \(p^2=2 a^2\)

c) \(a=2 p\)

d) \(p=2 a\)

17. Given that circle \(x^2+y^2-2 x+6 y+6=0\) and \(x^2+y^2-5 x+6 y+15=0\) touch, the equation to their common tangent is

a) \(x=3\)

b) \(y=6\)

c) \(7 x-12 y-21=0\)

d) \(7 x+12 y+21=0\)

18. The number of common tangents of the circles \(x^2+y^2-2 x-1=0\) and \(x^2+y^2-2 y-7=0\) is

a) 1

b) 2

c) 3

d) 4

19. A ray of light incident at the point \((-2,-1)\) gets reflected from the tangent at \((0,-1)\) to the circle \(x^2+y^2=1\). The reflected ray touches the circle. The equation of the line along which the incident ray moved is

a) \(4 x-3 y+11=0\)

b) \(4 x+3 y+11=0\)

c) \(3 x+4 y+11=0\)

d) None of these

20. If the points \(A(2,5)\) and \(B\) are symmetrical about the tangent to the circle \(x^2+y^2-4 x+4 y=0\) at the origin, then the coordinates of \(B\) are

a) \((5,-2)\)

b) \((1,5)\)

c) \((5,2)\)

d) None of these