Question 1 :
Consider a branch of the hyperbola \[ x ^ { 2 } - 2 y ^ { 2 } - 2 \sqrt { 2 } x - 4 \sqrt { 2 } y - 6 = 0 \]<br>with vertex at the point {tex} A {/tex}. Let {tex} B {/tex} be one of the end points of its latus rectum. If {tex} C {/tex} is the focus of the hyperbola nearest to the point {tex} A , {/tex} then the area of the triangle {tex} A B C {/tex} is<br>

Question 2 :
The centre of circle inscribed in square formed by the lines {tex} x ^ { 2 } - 8 x + 12 = 0 {/tex} and {tex} y ^ { 2 } - 14 y + 45 = 0 , {/tex} is

Question 3 :
Let {tex} x , y {/tex} be real variable satisfying the {tex} x ^ { 2 } + y ^ { 2 } + 8 x - 10 y - 40 = 0. {/tex} If {tex} a = \max \left\{ ( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } \right\} {/tex} and {tex} b = \min \left\{ ( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } \right\} , {/tex} then

Question 4 :
Suppose {tex} P {/tex} is a variable point on the hyperbola {tex} \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 {/tex} with eccentricity e. Let {tex} A ( \alpha , \beta ) {/tex} be a fixed point. The mid-point of {tex} A P {/tex} lies on

Question 5 :
If {tex} a , b > 0 , {/tex} then the angle of intersection of two parabolas {tex} y ^ { 2 } = a ^ { 3 } x {/tex} and {tex} x ^ { 2 } = b ^ { 3 } y {/tex} at a point other than the origin is

Question 6 :
The triangle {tex} P Q R {/tex} is inscribed in the circle {tex} x ^ { 2 } + y ^ { 2 } = 25 {/tex}. If {tex} Q {/tex} and {tex} R {/tex} have co-ordinates {tex} ( 3,4 ) {/tex} and {tex} ( - 4,3 ) {/tex} respectively, then {tex} \angle Q P R {/tex} is equal to

Question 7 :
The locus of the mid-point of a chord of the circle {tex} x ^ { 2 } + y ^ { 2 } = 4 {/tex} which subtends a right angle at the origin is

Question 8 :
If {tex}\mathrm P {/tex} is a point on the ellipse {tex}\mathrm { \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = }{/tex} with foci {tex}\mathrm S {/tex} and {tex}\mathrm S {/tex} and eccentricity {tex} \mathrm e , {/tex} then locus of the incentre of the triangle {tex}\mathrm { P S S ^ { \prime } }{/tex} is an ellipse of eccentricity

Question 9 :
<strong>Statement 1:</strong> If the parabola <em>y</em> = (<em>a</em>−<em>b</em>)<em>x</em>² + (<em>b</em>−<em>c</em>) + (<em>c</em> − <em>a</em>) touches the <em>x</em>-axis in the interval (0, 1), then the line ax + by + <em>c</em> = 0 always passes through a fixed point <strong><br> Statement 2:</strong> <p>The equation <em>L</em>₁ + <em>λ</em><em>L</em>₂ = 0 or <em>μ</em><em>L</em>₁ + <em>ν</em><em>L</em>₂ = 0 represent a line passing through the intersection of the lines <em>L</em>₁ = 0 and <em>L</em>₂ = 0</p> <p>Which is a fixed point, when <em>λ</em>, <em>μ</em>, <em>ν</em> are constants</p>

Question 10 :
Let length of transverse axis of a hyperbola with eccentricity {tex} \sqrt { 5 } {/tex} be {tex} 4 . {/tex} The difference between length of latus rectum and conjugate axis is

Question 11 :
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola {tex} y ^ { 2 } = 4 a x {/tex} is another parabola with directrix

Question 12 :
Let {tex} E {/tex} be the ellipse {tex} \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1 {/tex} and {tex} C {/tex} be the circle {tex} x ^ { 2 } + y ^ { 2 } = 9 . {/tex} Let {tex} P {/tex} and {tex} Q {/tex} be the points {tex} ( 1,2 ) {/tex} and {tex} ( 2,1 ) {/tex}, respectively, Then

Question 13 :
A square is inscribed in the circle {tex} x ^ { 2 } + y ^ { 2 } - 2 x + 4 y + 3 = 0 . {/tex} Its sides are parallel to the coordinate axes. The one vertex of the square is

Question 14 :
The radius of the circle passing through the foci of the ellipse {tex} \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1 , {/tex} and having its centre at {tex} ( 0,3 ) {/tex} is

Question 15 :
The normal {tex} y = m x - 2 a m - a m ^ { 3 } {/tex} to the parabola {tex} y ^ { 2 } = 4 a x {/tex} subtends a right angle at the vertex if

Question 16 :
The curve described parametrically by {tex} x = t ^ { 2 } + t + 1 {/tex}, {tex} y = t ^ { 2 } - t + 1 {/tex} represents

Question 17 :
Locus of intersection of the two perpendicular tangents to the given hyperbola is

Question 18 :
The line passing through the extremity {tex} A {/tex} of the major axis and extremity {tex} B {/tex} of the minor axis of the ellipse \[ x ^ { 2 } + 9 y ^ { 2 } = 9 \]<br>meets its auxiliary circle at the point {tex} M . {/tex} Then the area of the triangle with vertices at {tex} A , M {/tex} and the origin {tex} O {/tex} is<br>

Question 19 :
The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line {tex} 4 x - 5 y {/tex} {tex} = 20 {/tex} to the circle {tex} x ^ { 2 } + y ^ { 2 } = 9 {/tex} is

Question 20 :
A circle is given by {tex} x ^ { 2 } + ( y - 1 ) ^ { 2 } = 1 , {/tex} another circle {tex} C {/tex} touches it externally and also the {tex} x {/tex} -axis, then the locus of its centre is