Coordinate Geometry PYQ — Page 10

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___________

Suppose $A B$ is a focal chord of the parabola $y^2=12 x$ of length $l$ and slope $\mathrm{m} < \sqrt{3}$. If the distance of the chord $\mathrm{AB}$ from the origin is $\mathrm{d}$, then $l \mathrm{~d}^2$ is equal to _______

If the image of the point $(-4,5)$ in the line $x+2 y=2$ lies on the circle $(x+4)^2+(y-3)^2=r^2$, then $\mathrm{r}$ 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 $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$ 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 $R$ 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 :

If the line segment joining the points $(5,2)$ and $(2, a)$ subtends an angle $\frac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is :

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 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_________

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 _______.

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 $P$ 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 $(5,\frac{a}{4})$, 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 $R$ be a rectangle given by the lines $x=0,x=2$, $y=0$ and $y=5$. Let $A(\alpha ,0)$ and $B(0,\beta ),\alpha \in [0,2]$ and $\beta \in [0,5]$, be such that the line segment $AB$ divides the area of the rectangle $R$ in the ratio $4:1$. Then, the mid-point of $AB$ lies on a

The distance between the points (1, 2) and (4, 6) is:

Consider ellipses ${E}_{k}:k{x}^{2}+{k}^{2}{y}^{2}=1,k=1,2,\ldots ,20$. Let ${C}_{k}$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse ${E}_{k}$. If ${r}_{k}$ is the radius of the circle ${C}_{k}$, then the value of $\sum _{k=1}^{20}\frac{1}{{r}_{k}^{2}}$ is

Let $A(0,1),B(1,1)$ and $C(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola ${y}^{2}=4ax$ passing through $D$ is $(\alpha +\beta \sqrt{2},0)$, where $\alpha$ and $\beta$ are rational numbers, then $\frac{\alpha }{{\beta }^{2}}$ is equal to

The straight lines ${l}_{1}$ and ${l}_{2}$ pass through the origin and trisect the line segment of the line $L:9x+5y=45$ between the axes. If ${m}_{1}$ and ${m}_{2}$ are the slopes of the lines ${l}_{1}$ and ${l}_{2}$, then the point of intersection of the line $y=({m}_{1}+{m}_{2})x$ with $L$ lies on

The set of all values of ${a}^{2}$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $P(\frac{1+a}{2},\frac{1-a}{2})$ on the circle $2{x}^{2}+2{y}^{2}-(1+a)x-(1-a)y=0$, is equal to :

If the point $(\alpha ,\frac{7\sqrt{3}}{3})$ lies on the curve traced by the mid-points of the line segments of the lines $x\mathrm{cos}\theta +y\mathrm{sin}\theta =7,\theta \in (0,\frac{\pi }{2})$ between the co-ordinates axes, then $\alpha$ is equal to

Let the ellipse $E:{x}^{2}+9{y}^{2}=9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A,P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m-n$ is equal to

Let $y=x+2,4y=3x+6$ and $3y=4x+1$ be three tangent lines to the circle $(x-h{)}^{2}+(y-k{)}^{2}={r}^{2}$. Then $h+k$ is equal to :

A circle passing through the point $P(\alpha ,\beta )$ in the first quadrant touches the two coordinate axes at the points $A$ and $B.$ The point $P$ is above the line $AB.$ The point $Q$ on the line segment $AB$ is the foot of perpendicular from $P$ on $AB.$ If $PQ$ is equal to $11$ units, then the value of $\alpha \beta$ is $_______$

The parabolas : $a{x}^{2}+2bx+cy=0$ and ${d}^{2}+2ex+fy=0$ intersect on the line $y=1$. If $a,b,c,d,e,f$ are positive real numbers and $a,b,c$ are in $G.P.$, then