Question 1 :
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 2 :
The equation of the circle passing through {tex} ( 1,1 ) {/tex} and the points of intersection of {tex} x ^ { 2 } + y ^ { 2 } + 13 x - 3 y = 0 {/tex} and {tex} 2 x ^ { 2 } + 2 y ^ { 2 } + 4 x - 7 y - 25 = 0 {/tex} is
Question 3 :
Let {tex} A B {/tex} be a chord of the circle {tex} x ^ { 2 } + y ^ { 2 } = r ^ { 2 } {/tex} subtending a right angle at the centre. Then the locus of the centroid of the triangle {tex} PAB {/tex} as {tex}P{/tex} moves on the circle is
Question 4 :
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 5 :
An equation of the ellipse centered at ( 0,0 ) having eccentricity {tex} \frac { 3 } { 5 } {/tex} and passing through ( 4,0 ) is
Question 6 :
Eccentricity of a hyperbola angle between whose asymptotes is {tex} \frac { \pi } { 6 } {/tex} is
Question 7 :
The centre of the circle passing through the point {tex} ( 0,1 ) {/tex} and touching the curve {tex} y = x ^ { 2 } {/tex} at {tex} ( 2,4 ) {/tex} is
Question 8 :
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 9 :
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 10 :
The locus of the centre of a circle, which touches externally the circle {tex} x ^ { 2 } + y ^ { 2 } - 6 x - 6 y + 14 = 0 {/tex} and also touches the y-axis, is given by the equation:
Question 11 :
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 12 :
Let {tex} P , Q {/tex} and {tex} R {/tex} are three co-normal points on the parabola {tex} y ^ { 2 } = {/tex} {tex} 4 a x {/tex}. Then the correct statement(s) is/are
Question 13 :
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 14 :
The equation {tex} \frac { x ^ { 2 } } { 1 - r } - \frac { y ^ { 2 } } { 1 + r } = 1 , \quad r > 1 {/tex} represents
Question 15 :
If tangents are drawn to the ellipse {tex} x ^ { 2 } + 2 y ^ { 2 } = 2 , {/tex} then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is