Coordinate Geometry PYQ — Page 3

Let $\mathrm{A}(1,0), \mathrm{B}(2,-1)$ and $\mathrm{C}\left(\frac{7}{3}, \frac{4}{3}\right)$ be three points. If the equation of the bisector of the angle ABC is $\alpha x+\beta y=5$, then the value of $\alpha^{2}+\beta^{2}$ is

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 $\mathrm{S}=\left\{z \in \mathbb{C}:\left|\frac{z-6 i}{z-2 i}\right|=1\right.$ and $\left.\left|\frac{z-8+2 i}{z+2 i}\right|=\frac{3}{5}\right\}$. Then $\sum_{z \in \mathrm{~s}}|z|^{2}$ 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

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 $\mathrm{P}(10,2 \sqrt{15})$ 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 :

If the line $\alpha x+4 y=\sqrt{7}$, where $\alpha \in \mathbf{R}$, touches the ellipse $3 x^{2}+4 y^{2}=1$ at the point P in the first quadrant, then one of the focal distances of $P$ is :

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 $(h, k)$ 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 $\_\_\_\_$.

Let chord PQ of length $3\sqrt{13}$ of the parabola $y^2 = 12x$ be such that the ordinates of points $P$ and $Q$ are in the ratio $1:2$. If the chord PQ subtends an angle $\alpha$ at the focus of the parabola, then $\sin\alpha$ 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 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 the mid points of the sides of a triangle $ABC$ be $\left(\dfrac{5}{2}, 7\right)$, $\left(\dfrac{5}{2}, 3\right)$ and $(4, 5)$. If its incentre is $(h, k)$, then $3h + k$ is equal to :

Let $P$ be a moving point on the circle $x^2 + y^2 - 6x - 8y + 21 = 0$. Then, the maximum distance of $P$ from the vertex of the parabola $x^2 + 6x + y + 13 = 0$ is equal to:

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 $x = 9$ 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 the image of parabola $x^{2}=4 y$, in the line $x-y=1$ be $(y+a)^{2}=b(x-c)$, $a, b, c \in \mathrm{~N}$. Then $a+b+c$ 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 $A,B$ 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 ______

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: