Coordinate Geometry PYQ — Page 12

Let the equations of two adjacent sides of a parallelogram $ABCD$ be $2x-3y=-23$ and $5x+4y=23$. If the equation of its one diagonal $AC$ is $3x+7y=23$ and the distance of $A$ from the other diagonal is $d$, then $50{d}^{2}$ 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 ${H}_{n}:\frac{{x}^{2}}{1+n}-\frac{{y}^{2}}{3+n}=1,n\in \mathbb{N}$. Let $k$ be the smallest even value of $n$ such that the eccentricity of ${H}_{k}$ is a rational number. If $l$ is the length of the latus rectum of ${H}_{k}$, then $21l$ is equal to

A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio $2:3$, is a circle of radius

Let $(\alpha ,\beta )$ be the centroid of the triangle formed by the lines $15x-y=82,6x-5y=-4$ and $9x+4y=17$. Then $\alpha +2\beta$ and $2\alpha -\beta$ are the roots of the equation

Let $P({a}_{1},{b}_{1})$ and $Q({a}_{2},{b}_{2})$ be two distinct points on a circle with center $C(\sqrt{2},\sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$, then ${a}_{1}^{2}+{a}_{2}^{2}+{b}_{1}^{2}+{b}_{2}^{2}$ is equal to __________

Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{4}=1,a>2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $y$-axis, be $6\sqrt{3}$. Then the eccentricity of the ellipse is:

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

In an isosceles triangle $ABC$, the vertex $A$ is $(6,1)$ and the equation of the base $BC$ is $2x+y=4$. Let the point $B$ lie on the line $x+3y=7$. If $(\alpha ,\beta )$ is the centroid $\Delta ABC$, then $15(\alpha +\beta )$ is equal to

Let the area of the triangle with vertices $A(1,\alpha ),B(\alpha ,0)$ and $C(0,\alpha )$ be $4$ sq. units. If the points $(\alpha ,-\alpha ),(-\alpha ,\alpha )$ and $({\alpha }^{2},\beta )$ are collinear, then $\beta$ is equal to

Let $AB$ be a chord of length $12$ of the circle ${(x-2)}^{2}+{(y+1)}^{2}=\frac{169}{4}$ If tangents drawn to the circle at points $A$ and $B$ intersect at the point $P$, then five times the distance of point $P$ from chord $AB$ is equal to _____.

Let a triangle $ABC$ be inscribed in the circle ${x}^{2}-\sqrt{2}(x+y)+{y}^{2}=0$ such that $\angle BAC=\frac{\pi }{2}$. If the length of side $AB$ is $\sqrt{2}$, then the area of the $\triangle ABC$ is equal to:

If the length of the latus rectum of the ellipse ${x}^{2}+4{y}^{2}+2x+8y-\lambda =0$ is $4$, and $l$ is the length of its major axis, then $\lambda +l$ is equal to _____.

The eccentricity of the ellipse whose foci are (±2, 0) and length of latus rectum is 6 is:

A ray of light passing through the point $P(2,3)$ reflects on the $X$-axis at point $A$ and the reflected ray passes through the point $Q(5,4)$. Let $R$ be the point that divides the line segment $AQ$ internally into the ratio $2:1$. Let the co-ordinates of the foot of the perpendicular $M$ from $R$ on the bisector of the angle $PAQ$ be $(\alpha ,\beta )$. Then, the value of $7\alpha +3\beta$ is equal to _____.

Let the tangents at the points $P$ and $Q$ on the ellipse $\frac{{x}^{2}}{2}+\frac{{y}^{2}}{4}=1$ meet at the point $R(\sqrt{2},2\sqrt{2}-2)$. If $S$ is the focus of the ellipse on its negative major axis, then $S{P}^{2}+S{Q}^{2}$ is equal to

If one of the diameters of the circle ${x}^{2}+{y}^{2}-2\sqrt{2}x$ $-6\sqrt{2}y+14=0$ is a chord of the circle ${(x-2\sqrt{2})}^{2}$ $+{(y-2\sqrt{2})}^{2}={r}^{2}$, then the value of ${r}^{2}$ is equal to

An ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ passes through the vertices of the hyperbola $H:\frac{{x}^{2}}{49}-\frac{{y}^{2}}{64}=-1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113l$ is equal to _______.

Let $A(1,1),B(-4,3),C(-2,-5)$ be vertices of a triangle $ABC,P$ be a point on side $BC$, and ${\Delta }_{1}$ and ${\Delta }_{2}$ be the areas of triangle $APB$ and $ABC$. Respectively. If ${\Delta }_{1}:{\Delta }_{2}=4:7$, then the area enclosed by the lines $AP,AC$ and the $x$-axis is

The distance between the two points $A$ and ${A}^{'}$ which lie on $y=2$ such that both the line segments $AB$ and ${A}^{'}B$ (where $B$ is the point $(2,3)$) subtend angle $\frac{\pi }{4}$ at the origin, is equal to

Let $R$ be the point $(3,7)$ and let $P$ and $Q$ be two points on the line $x+y=5$ such that $PQR$ is an equilateral triangle. Then the area of $\Delta PQR$ is

Let the abscissae of the two points $P$ and $Q$ on a circle be the roots of ${x}^{2}-4x-6=0$ and the ordinates of $P$ and $Q$ be the roots of ${y}^{2}+2y-7=0$. If $PQ$ is a diameter of the circle ${x}^{2}+{y}^{2}+2ax+2by+c=0$, then the value of $(a+b-c)$ is

Let a circle $C$ touch the lines ${L}_{1}:4x-3y+{K}_{1}=0$ and ${L}_{2}:4x-3y+{K}_{2}=0,{K}_{1},{K}_{2}\in R$. If a line passing through the centre of the circle $C$ intersects ${L}_{1}$ at $(-1,2)$ and ${L}_{2}$ at $(3,-6)$, then the equation of the circle $C$ is

If the equation of the parabola, whose vertex is at $(5,4)$ and the directrix is $3x+y-29=0$, is ${x}^{2}+a{y}^{2}+bxy+cx+dy+k=0$, then $a+b+c+d+k$ is equal to