Question 1 :
The locus of the point of intersection of the tangents at the extremities of the chord of the ellipse {tex} x ^ { 2 } + 2 y ^ { 2 } {/tex} {tex} = 6 {/tex} which touches the ellipse {tex} x ^ { 2 } + 4 y ^ { 2 } = 4 {/tex} is
Question 2 :
Let {tex} \mathrm E _ { 1 } {/tex} be the ellipse {tex} \frac { x ^ { 2 } } { a ^ { 2 } + 2 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 {/tex} and {tex} \mathrm E _ { 2 } {/tex} be the ellipse {tex} \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } + 1 } = 1 . {/tex} The number of points from which to perpendicular tangents can be drawn to each of {tex}\mathrm E _ { 1 } {/tex} and {tex} \mathrm E _ { 2 } {/tex} is {tex} \ldots {/tex}
Question 3 :
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 4 :
An equation of a tangent common to the parabolas {tex} y ^ { 2 } = 4 x {/tex} and {tex} x ^ { 2 } = 4 y {/tex} is
Question 5 :
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 6 :
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 7 :
If {tex} x = 9 {/tex} is the chord of contact of the hyperbola {tex} x ^ { 2 } - y ^ { 2 } = 9 {/tex} then the equation of the corresponding pair of tangents is
Question 8 :
Let {tex} A {/tex} and {tex} B {/tex} are two points outside a circle {tex} S {/tex} such that the chord of contact from {tex} A {/tex} to {tex} S {/tex} passes through {tex} B {/tex}. If the length of tangent from {tex} A {/tex} to {tex} S {/tex} is {tex} l _ { 1 } {/tex} and length of tangent from {tex} B {/tex} to {tex} S {/tex} is {tex} l _ { 2 } , {/tex} then length of {tex} A B {/tex} is
Question 9 :
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 10 :
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 11 :
Let {tex} P Q {/tex} and {tex} R S {/tex} be tangents at the extremities of the diameter {tex} P R {/tex} of a circle of radius {tex} r {/tex}. If {tex} P S {/tex} and {tex} R Q {/tex} intersect at a point {tex} X {/tex} on the circumference of the circle, then {tex} 2 r {/tex} equals
Question 12 :
The line {tex} y = m x \pm \sqrt { a ^ { 2 } m ^ { 2 } - b ^ { 2 } } , m > 0 {/tex} touches the hyperbola {tex} \frac { x ^ { 2 } } { c ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 {/tex} at the point whose eccentric angle is
Question 13 :
<strong>Statement 1:</strong> If the parabola <em>y</em> = (<em>a</em>−<em>b</em>)<em>x</em><sup>2</sup> + (<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><sub>1</sub> + <em>λ</em><em>L</em><sub>2</sub> = 0 or <em>μ</em><em>L</em><sub>1</sub> + <em>ν</em><em>L</em><sub>2</sub> = 0 represent a line passing through the intersection of the lines <em>L</em><sub>1</sub> = 0 and <em>L</em><sub>2</sub> = 0</p> <p>Which is a fixed point, when <em>λ</em>, <em>μ</em>, <em>ν</em> are constants</p>
Question 14 :
Let {tex} P ( 3 \sec \theta , 2 \tan \theta ) {/tex} and {tex} Q ( 3 \sec \phi , 2 \tan \phi ) {/tex} where {tex} \theta + \phi = \frac { \pi } { 2 } , {/tex} be two distinct points on the hyperbola {tex} \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1 . {/tex} Then the ordinate of the point of intersection of the normals at {tex} P {/tex} and {tex} Q {/tex} is
Question 15 :
Each of the four inequalties given below defines a region in the {tex} x y {/tex} plane. One of these four regions does not have the following property. For any two points {tex} \left( x _ { 1 } , y _ { 1 } \right) {/tex} and {tex} \left( x _ { 2 } , y _ { 2 } \right) {/tex} in the region, the point {tex} \left( \frac { x _ { 1 } + x _ { 2 } } { 2 } , \frac { y _ { 1 } + y _ { 2 } } { 2 } \right) {/tex} is also in the region. The inequality defining this region is