7. On July 5 next, the Earth will reach aphelion, the point of its orbit where the distance from the Sun is maximum, approximately $1.52 \cdot 10 ^ { 11 } \mathrm {~m}$. Perihelion is instead the point at minimum distance from the Sun, approximately $1.47 \cdot 10 ^ { 11 } \mathrm {~m}$. Determine, in an appropriate coordinate system, the equation that represents the Earth's trajectory around the Sun.

7. If $P = ( x , y ) , F 1 = ( 3,0 ) , F 2 = ( - 3,0 )$ and $16 x 2 + 25 y 2 = 400$, then PF1 + PF2equals1 :
(A) 8
(B) 6
(C) 10
(D) 12

12. Let $\mathrm { P } ( \mathrm { a } \sec \mathrm { q } , \mathrm { b } \tan \mathrm { q } )$ and $\mathrm { Q } ( \mathrm { a } \sec \mathrm { q } , \mathrm { b } \tan \mathrm { q } )$ where $\mathrm { q } + \mathrm { q } = \pi / 2$, be two points on the hyperbola $\times 2 / \mathrm { a } 2 - \mathrm { y } 2 / \mathrm { b } 2 =$ 1. If ( $\mathrm { h } , \mathrm { k }$ ) is the point of intersection of the normals at P and Q , then K is equal to :
(A) $( a 2 + b 2 ) / a$
(B) $- ( ( a 2 + b 2 ) / a )$
(C) $( a 2 + b 2 ) / b$
(D) $- ( ( a 2 + b 2 ) / b )$

24. If $x = 9$ is the chord of contact of the hyperbola $x 2 - y 2 = 0$, then the equation of the corresponding pair of tangents is :
(A) $9 x 2 - 8 y 2 + 18 x - 9 = 0$
(B) $9 x 2 - 8 y 2 - 18 x + 9 = 0$
(C) $9 x 2 - 8 y 2 - 18 x - 9 = 0$
(D) $9 x 2 - 8 y 2 + 18 x + 9 = 0$

17. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y ^ { 2 } = 4 a x$ is another parabola with directrix
(A) $\mathrm { x } = - \mathrm { a }$
(B) $x = - a / 2$
(C) $\quad x = 0$
(D) $\quad x = a / 2$

11. For hyperbola $x ^ { 2 } / \left( \cos ^ { 2 } a \right) - y ^ { 2 } / \left( \sin ^ { 2 } a \right) = 1$ which of the following remains constant with change in ' a ' :
(a) abscissae of vertices
(b) abscissa of foci

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(c) eccentricity
(d) directrix

16. If a hyperbola passes through the focus of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ and its transverse and conjugate axes coincide with the major and minor axes of the ellipse, and the product of eccentricities is 1 , then
(A) the equation of hyperbola is $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$
(B) the equation of hyperbola is $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$
(C) focus of hyperbola is $( 5,0 )$
(D) focus of hyperbola is $( 5 \sqrt { 3 } , 0 )$

Sol. (A), (C)

Eccentricity of ellipse $= \frac { 3 } { 5 }$ Eccentricity of hyperbola $= \frac { 5 } { 3 }$ and it passes through $( \pm 3,0 )$ ⇒ its equation $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ where $1 + \frac { \mathrm { b } ^ { 2 } } { 9 } = \frac { 25 } { 9 } \Rightarrow \mathrm {~b} ^ { 2 } = 16$ $\Rightarrow \quad \frac { \mathrm { x } ^ { 2 } } { 9 } - \frac { \mathrm { y } ^ { 2 } } { 16 } = 1$ and its foci are $( \pm 5,0 )$.

A hyperbola, having the transverse axis of length $2\sin\theta$, is confocal with the ellipse $3x^2 + 4y^2 = 12$. Then its equation is
(A) $x^2\csc^2\theta - y^2\sec^2\theta = 1$
(B) $x^2\sec^2\theta - y^2\csc^2\theta = 1$
(C) $x^2\sin^2\theta - y^2\cos^2\theta = 1$
(D) $x^2\cos^2\theta - y^2\sin^2\theta = 1$

Consider a branch of the hyperbola
$$x ^ { 2 } - 2 y ^ { 2 } - 2 \sqrt { 2 } x - 4 \sqrt { 2 } y - 6 = 0$$
with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $A B C$ is
(A) $1 - \sqrt { \frac { 2 } { 3 } }$
(B) $\sqrt { \frac { 3 } { 2 } } - 1$
(C) $1 + \sqrt { \frac { 2 } { 3 } }$
(D) $\sqrt { \frac { 3 } { 2 } } + 1$

Match the conics in Column I with the statements/expressions in Column II.
Column I
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola
Column II
(p) The locus of the point $( h , k )$ for which the line $h x + k y = 1$ touches the circle $x ^ { 2 } + y ^ { 2 } = 4$
(q) Points $z$ in the complex plane satisfying $| z + 2 | - | z - 2 | = \pm 3$
(r) Points of the conic have parametric representation $x = \sqrt { 3 } \left( \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } } \right) , y = \frac { 2 t } { 1 + t ^ { 2 } }$
(s) The eccentricity of the conic lies in the interval $1 \leq x < \infty$
(t) Points $z$ in the complex plane satisfying $\operatorname { Re } ( z + 1 ) ^ { 2 } = | z | ^ { 2 } + 1$

The line $2 \mathrm { x } + \mathrm { y } = 1$ is tangent to the hyperbola $\frac { \mathrm { x } ^ { 2 } } { \mathrm { a } ^ { 2 } } - \frac { \mathrm { y } ^ { 2 } } { \mathrm {~b} ^ { 2 } } = 1$. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

Let $P ( 6,3 )$ be a point on the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the normal at the point $P$ intersects the $x$-axis at $( 9,0 )$, then the eccentricity of the hyperbola is
(A) $\sqrt { \frac { 5 } { 2 } }$
(B) $\sqrt { \frac { 3 } { 2 } }$
(C) $\sqrt { 2 }$
(D) $\sqrt { 3 }$

Consider the hyperbola $H : x ^ { 2 } - y ^ { 2 } = 1$ and a circle $S$ with center $N \left( x _ { 2 } , 0 \right)$. Suppose that $H$ and $S$ touch each other at a point $P \left( x _ { 1 } , y _ { 1 } \right)$ with $x _ { 1 } > 1$ and $y _ { 1 } > 0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If ( $l , m$ ) is the centroid of the triangle $\triangle P M N$, then the correct expression(s) is(are)
(A) $\frac { d l } { d x _ { 1 } } = 1 - \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$
(B) $\frac { d m } { d x _ { 1 } } = \frac { x _ { 1 } } { 3 \left( \sqrt { x _ { 1 } ^ { 2 } - 1 } \right) }$ for $x _ { 1 } > 1$
(C) $\frac { d l } { d x _ { 1 } } = 1 + \frac { 1 } { 3 x _ { 1 } ^ { 2 } }$ for $x _ { 1 } > 1$
(D) $\frac { d m } { d y _ { 1 } } = \frac { 1 } { 3 }$ for $y _ { 1 } > 0$

Let $F _ { 1 } \left( x _ { 1 } , 0 \right)$ and $F _ { 2 } \left( x _ { 2 } , 0 \right)$, for $x _ { 1 } < 0$ and $x _ { 2 } > 0$, be the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$. Suppose a parabola having vertex at the origin and focus at $F _ { 2 }$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.
The orthocentre of the triangle $F _ { 1 } M N$ is
(A) $\left( - \frac { 9 } { 10 } , 0 \right)$
(B) $\left( \frac { 2 } { 3 } , 0 \right)$
(C) $\left( \frac { 9 } { 10 } , 0 \right)$
(D) $\left( \frac { 2 } { 3 } , \sqrt { 6 } \right)$

Let $F _ { 1 } \left( x _ { 1 } , 0 \right)$ and $F _ { 2 } \left( x _ { 2 } , 0 \right)$, for $x _ { 1 } < 0$ and $x _ { 2 } > 0$, be the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$. Suppose a parabola having vertex at the origin and focus at $F _ { 2 }$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.
If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the $x$-axis at $Q$, then the ratio of area of the triangle $M Q R$ to area of the quadrilateral $M F _ { 1 } N F _ { 2 }$ is
(A) $3 : 4$
(B) $4 : 5$
(C) $5 : 8$
(D) $2 : 3$

If $2x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{16} = 1$, then which of the following CANNOT be sides of a right angled triangle?
[A] $a, 4, 1$
[B] $a, 4, 2$
[C] $2a, 8, 1$
[D] $2a, 4, 1$

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

	Column 1	Column 2	Column 3
	(I) $x^2 + y^2 = a^2$	(i) $my = m^2x + a$	(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)	$x^2 + a^2y^2 = a^2$	(ii) $y = mx + a\sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)	$y^2 = 4ax$	(iii) $y = mx + \sqrt{a^2m^2 - 1}$	(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)	$x^2 - a^2y^2 = a^2$	(iv) $y = mx + \sqrt{a^2m^2+1}$	(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

For $a = \sqrt{2}$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact $(-1, 1)$, then which of the following options is the only CORRECT combination for obtaining its equation?
[A] (I) (i) (P)
[B] (I) (ii) (Q)
[C] (II) (ii) (Q)
[D] (III) (i) (P)

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

	Column 1	Column 2	Column 3
	(I) $x^2 + y^2 = a^2$	(i) $my = m^2x + a$	(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)	$x^2 + a^2y^2 = a^2$	(ii) $y = mx + a\sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)	$y^2 = 4ax$	(iii) $y = mx + \sqrt{a^2m^2 - 1}$	(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)	$x^2 - a^2y^2 = a^2$	(iv) $y = mx + \sqrt{a^2m^2+1}$	(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

If a tangent to a suitable conic (Column 1) is found to be $y = x + 8$ and its point of contact is $(8, 16)$, then which of the following options is the only CORRECT combination?
[A] (I) (ii) (Q)
[B] (II) (iv) (R)
[C] (III) (i) (P)
[D] (III) (ii) (Q)

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

	Column 1	Column 2	Column 3
	(I) $x^2 + y^2 = a^2$	(i) $my = m^2x + a$	(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)	$x^2 + a^2y^2 = a^2$	(ii) $y = mx + a\sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)	$y^2 = 4ax$	(iii) $y = mx + \sqrt{a^2m^2 - 1}$	(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)	$x^2 - a^2y^2 = a^2$	(iv) $y = mx + \sqrt{a^2m^2+1}$	(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

The tangent to a suitable conic (Column 1) at $\left(\sqrt{3}, \frac{1}{2}\right)$ is found to be $\sqrt{3}x + 2y = 4$, then which of the following options is the only CORRECT combination?
[A] (IV) (iii) (S)
[B] (IV) (iv) (S)
[C] (II) (iii) (R)
[D] (II) (iv) (R)

Let $H$ : $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, where $a > b > 0$, be a hyperbola in the $x y$-plane whose conjugate axis $L M$ subtends an angle of $60 ^ { \circ }$ at one of its vertices $N$. Let the area of the triangle $L M N$ be $4 \sqrt { 3 }$.
LIST-I P. The length of the conjugate axis of $H$ is Q. The eccentricity of $H$ is R. The distance between the foci of $H$ is S. The length of the latus rectum of $H$ is
LIST-II

1. 8
2. $\frac { 4 } { \sqrt { 3 } }$
3. $\frac { 2 } { \sqrt { 3 } }$
4. 4

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(B) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 3 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 2 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 1 } ; \mathbf { R } \rightarrow \mathbf { 3 } ; \mathbf { S } \rightarrow \mathbf { 2 }$
(D) $\mathbf { P } \rightarrow \mathbf { 3 } ; \mathbf { Q } \rightarrow \mathbf { 4 } ; \mathbf { R } \rightarrow \mathbf { 2 } ; \mathbf { S } \rightarrow \mathbf { 1 }$

Define the collections $\left\{ E _ { 1 } , E _ { 2 } , E _ { 3 } , \ldots \right\}$ of ellipses and $\left\{ R _ { 1 } , R _ { 2 } , R _ { 3 } , \ldots \right\}$ of rectangles as follows: $E _ { 1 } : \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1 ;$ $R _ { 1 }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _ { 1 }$; $E _ { n }$ : ellipse $\frac { x ^ { 2 } } { a _ { n } ^ { 2 } } + \frac { y ^ { 2 } } { b _ { n } ^ { 2 } } = 1$ of largest area inscribed in $R _ { n - 1 } , n > 1$; $R _ { n }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _ { n } , n > 1$. Then which of the following options is/are correct?
(A) The eccentricities of $E _ { 18 }$ and $E _ { 19 }$ are NOT equal
(B) $\quad \sum _ { n = 1 } ^ { N } \left( \right.$ area of $\left. R _ { n } \right) < 24$, for each positive integer $N$
(C) The length of latus rectum of $E _ { 9 }$ is $\frac { 1 } { 6 }$
(D) The distance of a focus from the centre in $E _ { 9 }$ is $\frac { \sqrt { 5 } } { 32 }$

For the hyperbola $\frac { x ^ { 2 } } { \cos ^ { 2 } \alpha } - \frac { y ^ { 2 } } { \sin ^ { 2 } \alpha } = 1$, which of the following remains constant when $\alpha$ varies?
(1) eccentricity
(2) directrix
(3) abscissae of vertices
(4) abscissae of foci

If ( $2,3,5$ ) is one end of a diameter of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x - 12 y - 2 z + 20 = 0$, then the coordinates of the other end of the diameter are
(1) $( 4,9 , - 3 )$
(2) $( 4 , - 3,3 )$
(3) $( 4,3,5 )$
(4) $( 4,3 , - 3 )$

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$ is
(1) $5x^{2}+3y^{2}-48=0$
(2) $3x^{2}+5y^{2}-15=0$
(3) $5x^{2}+3y^{2}-32=0$
(4) $3x^{2}+5y^{2}-32=0$

In a $\triangle PQR$, if $3\sin P + 4\cos Q = 6$ and $4\sin Q + 3\cos P = 1$, then the angle $R$ is equal to
(1) $\frac{5\pi}{6}$
(2) $\frac{\pi}{6}$
(3) $\frac{\pi}{4}$
(4) $\frac{3\pi}{4}$