Coordinate Geometry PYQ — Page 12

Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2x=2$ such that $\Delta ABC$ is an equilateral triangle. Then, the area of the $\Delta ABC$ is

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 $H$ be the hyperbola, whose foci are $(1\pm \sqrt{2},0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is:

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

Consider the triangles with vertices $A(2,1),B(0,0)$ and $C(t,4),t=[0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha +21\beta$ is equal to ___________.

Let $PQ$ be a focal chord of the parabola ${y}^{2}=36x$ of length $100,$ making an acute angle with the positive $x-$axis. Let the ordinate of $P$ be positive and $M$ be the point on the line segment $PQ$ such that $PM:MQ=3:1.$ Then which of the following points does $\mathrm{NOT}$ lie on the line passing through M and perpendicular to the line $PQ$?

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

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 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 _____.

Let the tangent to the circle ${C}_{1}:{x}^{2}+{y}^{2}=2$ at the point $M(-1,1)$ intersect the circle ${C}_{2}$ : ${(x-3)}^{2}+{(y-2)}^{2}=5$, at two distinct points $A$ and $B$. If the tangents to ${C}_{2}$ at the points $A$ and $B$ intersect at $N$, then the area of the triangle $ANB$ is equal to

Let the tangents at two points $A$ and $B$ on the circle ${x}^{2}+{y}^{2}-4x+3=0$ meet at origin $O(0,0)$. Then the area of the triangle of $OAB$ 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 _____.

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

Let $S={(x,y)\in \mathbb{N}\times \mathbb{N}:9{(x-3)}^{2}+16{(y-4)}^{2}\leq 144}$ and $T={(x,y)\in \mathbb{R}\times \mathbb{R}:{(x-7)}^{2}+{(y-4)}^{2}\leq 36}$ The $n(S\cap T)$ is equal to ______.

Let a circle $C:{(x-h)}^{2}+{(y-k)}^{2}={r}^{2},k>0$, touch the $x$-axis at $(1,0)$. If the line $x+y=0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is $2$, then the value of $h+k+r$ is equal to _____.

Let the eccentricity of an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$, be $\frac{1}{4}$. If this ellipse passes through the point $(-4\sqrt{\frac{2}{5}},3)$, then ${a}^{2}+{b}^{2}$ 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 eccentricity of the ellipse whose foci are (±2, 0) and length of latus rectum is 6 is:

The distance of the origin from the centroid of the triangle whose two sides have the equations $x-2y+1=0$ and $2x-y-1=0$ and whose orthocenter is $(\frac{7}{3},\frac{7}{3})$ is:

If the circles ${x}^{2}+{y}^{2}+6x+8y+16=0$ and ${x}^{2}+{y}^{2}+2(3-\sqrt{3})x+2(4-\sqrt{6})y=k+6\sqrt{3}+8\sqrt{6}$, $k>0$, touch internally at the point $P(\alpha ,\beta )$, then ${(\alpha +\sqrt{3})}^{2}+{(\beta +\sqrt{6})}^{2}$ 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

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