Coordinate Geometry PYQ — Page 8

Let the centre of a circle, passing through the points $(0,0),(1,0)$ and touching the circle $x^2+y^2=9$, be $(h, k)$. Then for all possible values of the coordinates of the centre $(h, k), 4\left(h^2+k^2\right)$ is equal to_________

Let PQ be a chord of the parabola $y^2=12 x$ and the midpoint of PQ be at $(4,1)$. Then, which of the following point lies on the line passing through the points $\mathrm{P}$ and Q?

Let $A$ be the point of intersection of the lines $3x+2y=14,5x-y=6$ and $B$ be the point of intersection of the lines $4x+3y=8,6x+y=5$. The distance of the point $P(5,-2)$ from the line $AB$ is

Let $R$ be the interior region between the lines $3x-y+1=0$ and $x+2y-5=0$ containing the origin. The set of all values of $a$, for which the points $({a}^{2},a+1)$ lie in $R$, is :

Let $H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4 \sqrt{3}$. Suppose the point $(\alpha, 6), \alpha>0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$, then $\alpha^2+\beta$ is equal to

Let $\mathrm{S}$ be the focus of the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$, on the positive $x$-axis. Let $\mathrm{C}$ be the circle with its centre at $A(\sqrt{6}, \sqrt{5})$ and passing through the point $S$. If $O$ is the origin and $S A B$ is a diameter of $C$, then the square of the area of the triangle OSB is equal to___________

Let ${e}_{1}$ be the eccentricity of the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{9}=1$ and ${e}_{2}$ be the eccentricity of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$, which passes through the foci of the hyperbola. If ${e}_{1}{e}_{2}=1$, then the length of the chord of the ellipse parallel to the $x$-axis and passing through $(0,2)$ is :

Let $(5,\frac{a}{4})$, be the circumcenter of a triangle with vertices $A(a,-2),B(a,6)$ and $C(\frac{a}{4},-2)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha +\beta +\gamma$ is

Let $C$ be the circle of minimum area touching the parabola $y=6-x^2$ and the lines $y=\sqrt{3}|x|$. Then, which one of the following points lies on the circle $C$ ?

Let $ABC$ be an isosceles triangle in which $A$ is at $(-1,0),\angle A=\frac{2\pi }{3},AB=AC$ and $B$ is on the positive $x-$axis. If $BC=4\sqrt{3}$ and the line $BC$ intersects the line $y=x+3$ at $(\alpha ,\beta ),$ then $\frac{{\beta }^{4}}{{\alpha }^{2}}$ is:

Let $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ is:

Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is -3 . If the points $(x, y)$ satisfying $\mathrm{A}^2+x \mathrm{~A}+y \mathrm{I}=\mathrm{O}$ lie on a hyperbola, whose $\text { length of semi major axis is } x \text { and semi minor axis is } y$, eccentricity is $\mathrm{e}$ and the length of the latus rectum is $l$, then $81\left(e^ 4+l^2\right)$ is equal to

Let $P(\alpha ,\beta )$ be a point on the parabola ${y}^{2}=4x$. If $P$ also lies on the chord of the parabola ${x}^{2}=8y$ whose mid point is $(1,\frac{5}{4})$, then $(\alpha -28)(\beta -8)$ is equal to _______.

Let $P$ be a point on the hyperbola $H:\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}=1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then, the square of the distance of $P$ from the origin is

Let $P$ be a point on the ellipse $\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1$. Let the line passing through $P$ and parallel to $y-$axis meet the circle ${x}^{2}+{y}^{2}=9$ at point $Q$ such that $P\text{and}Q$ are on the same side of the $x-$axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR:RQ=4:3$ as $P$ moves on the ellipse, is:

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$ of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$, is

Let $C$ be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersects the circle $\mathrm{C}$ at the points $\mathrm{P}$ and $\mathrm{Q}$. Let $\mathrm{MN}$ be a chord of $\mathrm{C}$ of length 2 unit and slope -1. Then, a distance (in units) between the chord PQ and the chord $\mathrm{MN}$ is

Let $A(a,b),B(3,4)$ and $(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3,7b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is

Let $\alpha ,\beta ,\gamma ,\delta \in Z$ and let $A(\alpha ,\beta ),B(1,0),C(\gamma ,\delta )$ and $D(1,2)$ be the vertices of a parallelogram $ABCD$. If $AB=\sqrt{10}$ and the points $A$ and $C$ lie on the line $3y=2x+1$, then $2(\alpha +\beta +\gamma +\delta )$ is equal to

Let $f(x)=x^2+9, g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f \circ g(10), \mathrm{b}=g \circ f(3)$. If $\mathrm{e}$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$, then $8 \mathrm{e}^2+l^2$ is equal to.

Let $A(-1,1)$ and $B(2,3)$ be two points and $P$ be a variable point above the line $A B$ such that the area of $\triangle \mathrm{PAB}$ is 10 . If the locus of $\mathrm{P}$ is $\mathrm{a} x+\mathrm{b} y=15$, then $5 \mathrm{a}+2 \mathrm{~b}$ is :

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is $56$ more than the total number of subsets of $B$. Then the distance of the point $P(m,n)$ from the point $Q(-2,-3)$ is

Let $C:{x}^{2}+{y}^{2}=4$ and ${C}^{'}:{x}^{2}+{y}^{2}-4\lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $C$ and $C'$ intersect at two distinct points, is $R-[a,b]$, then the point $(8a+12,16b-20)$ lies on the curve:

Let $A, B$ and $C$ be three points on the parabola $y^2=6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left(\frac{A M \cdot B N}{C D}\right)^2$ is equal to _________