Coordinate Geometry PYQ — Page 8

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha^2+2 \beta^2$ is equal to

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

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5: 4$, is $a x^2+b y^2+c x y+d x+e y+170=0$, then the value of $a^2+2 b+3 c+4 d+e$ is equal to :

A ray of light coming from the point $P(1,2)$ gets reflected from the point $Q$ on the $x$-axis and then passes through the point $R(4,3)$. If the point $S(h, k)$ is such that PQRS is a parallelogram, then $h k^2$ is equal to :

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 the locus of the mid points of the chords of circle ${x}^{2}+{(y-1)}^{2}=1$ drawn from the origin intersect the line $x+y=1$ at $P$ and $Q$. Then, the length of $PQ$ is:

If the sum of squares of all real values of $\alpha$, for which the lines $2x-y+3=0,6x+3y+1=0$ and $\alpha x+2y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is ________.

Let a circle passing through $(2,0)$ have its centre at the point $(h, k)$. Let $\left(x_c, y_c\right)$ be the point of intersection of the lines $3 x+5 y=1$ and $(2+c) x+5 c^2 y=1$. If $\mathrm{h}=\lim _{\mathrm{c} \rightarrow 1} x_{\mathrm{c}}$ and $\mathrm{k}=\lim _{\mathrm{c} \rightarrow 1} y_{\mathrm{c}}$, then the equation of the circle is :

If the shortest distance of the parabola ${y}^{2}=4x$ from the centre of the circle ${x}^{2}+{y}^{2}-4x-16y+64=0$ is $d$, then ${d}^{2}$ 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 the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius 2 . If the line of the common chord of $\mathrm{C}_1$ and $\mathrm{C}_2$ intersects the $y$-axis at the point $\mathrm{P}$, then the square of the distance of $\mathrm{P}$ from the centre of $\mathrm{C}_1$ is :

Consider a circle $(x-{\alpha }^{2})+(y-{\beta }^{2})=50$, where $\alpha ,\beta >0$. If the circle touches the line $y+x=0$ at the point $P$, whose distance from the origin is $4\sqrt{2}$ , then $(\alpha +\beta {)}^{2}$ is equal to _______.

The length of the chord of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1$, whose mid point is $(1,\frac{2}{5})$, is equal to:

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 the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2: 1$, then $(\alpha+\beta)+4\left(r_1^2+r_2^2\right)$ equals

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 $\alpha ,\beta ,\gamma ,\delta \in Z$ 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

The vertices of a triangle are $\mathrm{A}(-1,3), \mathrm{B}(-2,2)$ and $\mathrm{C}(3,-1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

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 the line $L:\sqrt{2}x+y=\alpha$ pass through the point of the intersection $P$(in the first quadrant)of the circle ${x}^{2}+{y}^{2}=3$ and the parabola ${x}^{2}=2y$. Let the line $L$ touch two circles ${C}_{1}$ and ${C}_{2}$ of equal radius $2\sqrt{3}$. If the centres ${Q}_{1}$ and ${Q}_{2}$ of the circles ${C}_{1}$ and ${C}_{2}$ lie on the $y-$axis, then the square of the area of the triangle $P{Q}_{1}{Q}_{2}$ is equal to _________.

A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m+n^2$ is equal to

Let $A B C D$ and $A E F G$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment $\mathrm{AB}$ and the point $\mathrm{F}$ is on the diagonal $\mathrm{AC}$. Then the radius $\mathrm{r}$ of the circle passing through the point $\mathrm{F}$ and touching the line segments $\mathrm{BC}$ and $\mathrm{CD}$ satisfies:

Equations of two diameters of a circle are $2x-3y=5$ and $3x-4y=7$. The line joining the points $(-\frac{22}{7},-4)$ and $(-\frac{1}{7},3)$ intersects the circle at only one point $P(\alpha ,\beta )$. Then $17\beta -\alpha$ is equal to

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :