Coordinate Geometry PYQ — Page 2

Let PQ be a chord of the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{b^{2}}=1$, perpendicular to the x -axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$, then the area of the triangle OPQ is

Let $x = 9$ be a directrix of an ellipse $E$, whose centre is at the origin and eccentricity is $\dfrac{1}{3}$. Let $P(\alpha, 0)$, $\alpha > 0$, be a focus of $E$ and $AB$ be a chord passing through $P$. Then the locus of the mid point of $AB$ is :

Let S and $\mathrm{S}^{\prime}$ be the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ and $\mathrm{P}(\alpha, \beta)$ be a point on the ellipse in the first quadrant. If $(\mathrm{SP})^{2}+\left(\mathrm{S}^{\prime} \mathrm{P}\right)^{2}-\mathrm{SP} \cdot \mathrm{S}^{\prime} \mathrm{P}=37$, then $\alpha^{2}+\beta^{2}$ is equal to :

If the chord joining the points $\mathrm{P}_{1}\left(x_{1}, y_{1}\right)$ and $\mathrm{P}_{2}\left(x_{2}, y_{2}\right)$ on the parabola $y^{2}=12 x$ subtends a right angle at the vertex of the parabola, then $x_{1} x_{2}-y_{1} y_{2}$ is equal to

For some $\theta \in\left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^{2}-y^{2} \sec ^{2} \theta=8$ be $e_{1}$ and $l_{1}$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^{2} \sec ^{2} \theta+y^{2}=6$ be $e_{2}$ and $l_{2}$, respectively. If $e_{1}^{2}=e_{2}^{2}\left(\sec ^{2} \theta+1\right)$, then $\left(\frac{l_{1} l_{2}}{e_{1} e_{2}}\right) \tan ^{2} \theta$ is equal to $\_\_\_\_$

Let a circle of radius 4 pass through the origin O, the points $\mathrm{A}(-\sqrt{3} a, 0)$ and $\mathrm{B}(0,-\sqrt{2} b)$, where $a$ and $b$ are real parameters and $a b \neq 0$. Then the locus of the centroid of $\triangle O A B$ is a circle of radius

Let the line $L_1 : x + 3 = 0$ intersect the lines $L_2 : x - y = 0$ and $L_3 : 3x + y = 0$ at the points $A$ and $B$, respectively. Let the bisector of the obtuse angle between the lines $L_2$ and $L_3$ intersect the line $L_1$ at the point $C$. Then $BC^2 : AC^2$ is equal to:

Let O be the vertex of the parabola $x^{2}=4 y$ and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio $2: 3$ be the conic C. Then the equation of the chord of $C$, which is bisected at the point $(1,2)$, is :

Let $y^{2}=12 x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the $x$-axis such that $\angle \mathrm{OPA}=90^{\circ}$. Then the locus of the centroid of such triangles OPA is :

If the points of intersection of the ellipses $x^{2}+2 y^{2}-6 x-12 y+23=0$ and $4 x^{2}+2 y^{2}-20 x-12 y+35=0$ lie on a circle of radius $r$ and centre $(a, b)$, then the value of $a b+18 r^{2}$ is

Let each of the two ellipses $\mathrm{E}_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,(a>b)$ and $\mathrm{E}_{2}: \frac{x^{2}}{\mathrm{~A}^{2}}+\frac{y^{2}}{\mathrm{~B}^{2}}=1,(\mathrm{~A}<\mathrm{B})$ have eccentricity $\frac{4}{5}$. Let the lengths of the latus recta of $E_{1}$ and $E_{2}$ be $l_{1}$ and $l_{2}$, respectively, such that $2 l_{1}^{2}=9 l_{2}$. If the distance between the foci of $E_{1}$ is 8, then the distance between the foci of $E_{2}$ is

Let the centre of the circle $x^2 + y^2 + 2gx + 2fy + 25 = 0$ be in the first quadrant and lie on the line $2x - y = 4$. Let the area of an equilateral triangle inscribed in the circle be $27\sqrt{3}$. Then the square of the length of the chord of the circle on the line $x = 1$ is _______.

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x + (k-1)y + 3 = 0$ and $2x + k^2 y - 4 = 0$. If the line $x - y + 2 = 0$ intersects the circle at the points A and B, then $(AB)^2$ is equal to:

The eccentricity of an ellipse $E$ with centre at the origin $O$ is $\dfrac{\sqrt{3}}{2}$ and its directrices are $x = \pm \dfrac{4\sqrt{6}}{3}$. Let $H: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of $E$, and whose length of latus rectum is equal to the length of minor axis of $E$. Then the distance between the foci of $H$ is :

Let PQ and MN be two straight lines touching the circle $x^{2}+y^{2}-4 x-6 y-3=0$ at the points A and B respectively. Let O be the centre of the circle and $\angle \mathrm{AOB}=\pi / 3$. Then the locus of the point of intersection of the lines PQ and MN is :

An equilateral triangle OAB is inscribed in the parabola $y^{2}=4 x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having $A B$ as a diameter from the origin is

Let the locus of the mid-point of the chord through the origin O of the parabola $y^{2}=4 x$ be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio $3: 1$, is :

If the line $\alpha x+2 y=1$, where $\alpha \in \mathbb{R}$, does not meet the hyperbola $x^{2}-9 y^{2}=9$, then a possible value of $\alpha$ is:

Let the length of the latus rectum of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,(a>b)$, be 30. If its eccentricity is the maximum value of the function $f(t)=-\frac{3}{4}+2 t-t^{2}$, then ($a^{2}+b^{2}$) is equal to

Let the set of all values of $r$, for which the circles $(x+1)^{2}+(y+4)^{2}=r^{2}$ and $x^{2}+y^{2}-4 x-2 y-4=0$ intersect at two distinct points be the interval $(\alpha, \beta)$. Then $\alpha \beta$ is equal to

Let $P(3\cos\alpha, 2\sin\alpha)$, $\alpha \neq 0$, be a point on the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$, $Q$ be a point on the circle $x^2+y^2-14x-14y+82=0$ and $R$ be a point on the line $x+y=5$ such that the centroid of the triangle $PQR$ is $\left(2+\cos\alpha, 3+\dfrac{2}{3}\sin\alpha\right)$. Then the sum of the ordinates of all possible points $R$ is:

Let the foci of a hyperbola coincide with the foci of the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is :

Let $\mathrm{A}(1,2)$ and $\mathrm{C}(-3,-6)$ be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line $7 x-y=14$. If $\mathrm{B}(\alpha, \beta)$ and $\mathrm{D}(\gamma, \delta)$ are the other two vertices, then $|\alpha+\beta+\gamma+\delta|$ is equal to

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $x+2 \sqrt{2} y=4$. If the co-ordinates of the vertex A are $(\alpha, \beta)$, then the greatest integer less than or equal to $|\alpha+\sqrt{2} \beta|$ is