Coordinate Geometry PYQ — Page 3

Consider the circle $C: x^2+y^2-6x-8y-11=0$. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle $x^2+y^2-\alpha x - \beta y - \gamma = 0$, then $\alpha + \beta + 2\gamma$ is equal to ________.

Let one end of a focal chord of the parabola $y^{2}=16 x$ be $(16,16)$. If $\mathrm{P}(\alpha, \beta)$ divides this focal chord internally in the ratio $5: 2$, then the minimum value of $\alpha+\beta$ is equal to :

Let the directrix of the parabola $P: y^2 = 8x$, cut $x$-axis at the point $A$. Let $B(\alpha, \beta)$, $\alpha > 1$, be a point on $P$ such that the slope of $AB$ is $3/5$. If $BC$ is a focal chord of $P$, then six times the area of $\triangle ABC$ is :

Let $A,B$ be points on the two half-lines $x-\sqrt{3}|y|=\alpha$, $\alpha>0$ at a distance of $\alpha$ from their point of intersection $P$. The line segment $AB$ meets the angle bisector of the given half-lines at the point $Q$. If $PQ=\dfrac{9}{2}$ and $R$ is the radius of the circumcircle of $\triangle PAB$, then $\dfrac{\alpha^2}{R}$ is equal to ______

Let $(h, k)$ lie on the circle $\mathrm{C}: x^{2}+y^{2}=4$ and the point $(2 h+1,3 k+2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^{2}}$ is equal to $\_\_\_\_$.

If a straight line drawn through the point of intersection of the lines $4x+3y-1=0$ and $3x+4y-1=0$, meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is:

If the eccentricity $e$ of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, passing through $(6, 4\sqrt{3})$, satisfies $15(e^2 + 1) = 34e$, then the length of the latus rectum of the hyperbola $\dfrac{x^2}{b^2} - \dfrac{y^2}{2(a^2+1)} = 1$ is:

Let $A,B$ and $C$ be the vertices of a variable right angled triangle inscribed in the parabola $y^2=16x$. Let the vertex $B$ containing the right angle be $(4,8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_o$. Then three times the length of latus rectum of $C_o$ is ______

Let $\mathrm{P}(10,2 \sqrt{15})$ be a point on the hyperbola $\frac{x^{2}}{\mathrm{a}^{2}}-\frac{y^{2}}{\mathrm{~b}^{2}}=1$, whose foci are S and $\mathrm{S}^{\prime}$. If the length of its latus rectum is 8, then the square of the area of $\Delta \mathrm{PSS}^{\prime}$ is equal to :

Let O be the vertex of the parabola $y^2=4x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is:

Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is $e$, then $72e^2$ is equal to _______.

Let the parabola $y = x^2 + px + q$ passing through the point $(1, -1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2 + q^2$ is:

Let $H: \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be a hyperbola such that the distance between its foci is $6$ and the distance between its directrices is $\dfrac{8}{3}$. If the line $x=\alpha$ intersects the hyperbola $H$ at the points $A$ and $B$ such that the area of the triangle $AOB$ is $4\sqrt{15}$, where $O$ is the origin, then $\alpha^2$ equals

Let an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a < b$, pass through the point $(4, 3)$ and have eccentricity $\dfrac{\sqrt{5}}{3}$. Then the length of its latus rectum is :

An ellipse has its center at $(1,-2)$, one focus at $(3,-2)$ and one vertex at $(5,-2)$. Then the length of its latus rectum is :

Among the statements $(S 1)$ : If $A(5,-1)$ and $B(-2,3)$ are two vertices of a triangle, whose orthocentre is $(0,0)$, then its third vertex is $(-4,-7)$ and (S2) : If positive numbers 2a, b, c are three consecutive terms of an A.P., then the lines $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ are concurrent at $(2,-2)$,

Let the point $P$ be the vertex of the parabola $y = x^2 - 6x + 12$. If a line passing through the point $P$ intersects the circle $x^2 + y^2 - 2x - 4y + 3 = 0$ at the points $R$ and $S$, then the maximum value of $(PR + PS)^2$ is :

Let a circle $C$ have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of $C$ on the line $x + y = 1$ is $\sqrt{14}$, then the square of the radius of $C$ is _______.

Consider the parabola $P : y^2 = 4kx$ and the ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of $P$ and $E$, be their latus rectums. If the eccentricity of $E$ is $e$, then $e^2 + 2\sqrt{2}$ is equal to _____.

Let a focus of the ellipse $E: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ be $S(4, 0)$ and its eccentricity be $\dfrac{4}{5}$. If the point $P(3, \alpha)$ lies on $E$ and $O$ is the origin, then the area of $\triangle POS$ is equal to:

Let the angles made with the positive $x$-axis by two straight lines drawn from the point $\mathrm{P}(2,3)$ and meeting the line $x+y=6$ at a distance $\sqrt{\frac{2}{3}}$ from the point P be $\theta_{1}$ and $\theta_{2}$. Then the value of $\left(\theta_{1}+\theta_{2}\right)$ is:

Let the eccentricity $e$ of a hyperbola satisfy the equation $6e^2 - 11e + 3 = 0$. If the foci of the hyperbola are $(3, 5)$ and $(3, -4)$, then the length of its latus rectum is :

Let $C$ be a circle having centre in the first quadrant and touching the $x$-axis at a distance of $3$ units from the origin. If the circle $C$ has an intercept of length $6\sqrt{3}$ on $y$-axis, then the length of the chord of the circle $C$ on the line $x - y = 3$ is :

Let the line $x - y = 4$ intersect the circle $C: (x-4)^2 + (y+3)^2 = 9$ at the points $Q$ and $R$. If $P(\alpha, \beta)$ is a point on $C$ such that $PQ = PR$, then $(6\alpha + 8\beta)^2$ is equal to __________.