Coordinate Geometry PYQ — Page 13

Let ${P}_{1}$ be a parabola with vertex $(3,2)$ and focus $(4,4)$ and ${P}_{2}$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of ${P}_{2}$ is $x+2y=$ _____.

The equation of a circle with center (2, 3) and radius 5 is:

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 $a>0,b>0$. Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$. Let ${e}^{'}$ and ${l}^{'}$ respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If ${e}^{2}=\frac{11}{14}l$ and ${({e}^{'})}^{2}=\frac{11}{8}{l}^{'}$, then the value of $77a+44b$ is equal to

Let the mirror image of a circle ${c}_{1}:{x}^{2}+{y}^{2}-2x-6y+\alpha =0$ in line $y=x+1$ be ${c}_{2}:5{x}^{2}+5{y}^{2}+10gx$ $+10fy+38=0$. If $r$ is the radius of circle ${c}_{2}$, then $\alpha +6{r}^{2}$ is equal to ______

For $t\in (0,2\pi )$, if $ABC$ is an equilateral triangle with vertices $A(\mathrm{sin}t,-\mathrm{cos}t),B(\mathrm{cos}t,\mathrm{sin}t)$ and $C(a,b)$ such that its orthocentre lies on a circle with centre $(1,\frac{1}{3})$, then $({a}^{2}-{b}^{2})$ is equal to

The equations of the sides $AB,BC$ and $CA$ of a triangle $ABC$ are $2x+y=0,x+py=15a$ and $x-y=3$ respectively. If its orthocentre is $(2,a)$, $-\frac{1}{2}<a<2$, then $p$ is equal to

Let $H:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1,a>0,b>0$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2\sqrt{2}+\sqrt{14})$. If the eccentricity $H$ is $\frac{\sqrt{11}}{2}$, then value of ${a}^{2}+{b}^{2}$ is equal to ______.

If the length of the latus rectum of a parabola, whose focus is $(a,a)$ and the tangent at its vertex is $x+y=a$, is $16$, then $|a|$ is equal to

Let ${m}_{1},{m}_{2}$ be the slopes of two adjacent sides of a square of side $a$ such that ${a}^{2}+11a+3({m}_{1}^{2}+{m}_{2}^{2})=220$. If one vertex of the square is $(10(\mathrm{cos}\alpha -\mathrm{sin}\alpha ),10(\mathrm{sin}\alpha +\mathrm{cos}\alpha ))$, where $\alpha \in (0,\frac{\pi }{2})$ and the equation of one diagonal is $(cos\alpha -sin\alpha )x+(\mathrm{sin}\alpha +\mathrm{cos}\alpha )y=10$, then $72({\mathrm{sin}}^{4}\alpha +{\mathrm{cos}}^{4}\alpha )+{a}^{2}-3a+13$ is equal to

Let the hyperbola $H:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ pass through the point $(2\sqrt{2},-2\sqrt{2})$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is e times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?

Let a circle $C$ of radius $5$ lie below the $x$-axis. The line ${L}_{1}=4x+3y+2$ passes through the centre $P$ of the circle $C$ and intersects the line ${L}_{2}:3x-4y-11=0$ at $Q$. The line ${L}_{2}$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5x-12y+51=0$ is

A circle touches both the $y$-axis and the line $x+y=0$. Then the locus of its center

A circle ${C}_{1}$ passes through the origin $O$ and has diameter $4$ on the positive $x$-axis. The line $y=2x$ gives a chord $OA$ of a circle ${C}_{1}$. Let ${C}_{2}$ be the circle with $OA$ as a diameter. If the tangent to ${C}_{2}$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $QA:AP$ is equal to

The set of values of $k$ for which the circle $C:4{x}^{2}+4{y}^{2}-12x+8y+k=0$ lies inside the fourth quadrant and the point $(1,-\frac{1}{3})$ lies on or inside the circle $C$ is

Let the abscissae of the two points $P$ and $Q$ be the roots of $2{x}^{2}-rx+p=0$ and the ordinates of $P$ and $Q$ be the roots of ${x}^{2}-sx-q=0$. If the equation of the circle described on $PQ$ as diameter is $2({x}^{2}+{y}^{2})-11x-14y-22=0$, then $2r+s-2q+p$ is equal to ______.

Let $x=2t,y=\frac{{t}^{2}}{3}$ be a conic. Let $S$ be the focus and $B$ be the point on the axis of the conic such that $SA\perp BA$, where $A$ is any point on the conic. If $k$ is the ordinate of the centroid of the $\Delta SAB$, then $\underset{t\rightarrow 1}{\mathrm{lim}}k$ is equal to

The locus of the mid-point of the line segment joining the point $(4,3)$ and the points on the ellipse ${x}^{2}+2{y}^{2}=4$ is an ellipse with eccentricity

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 vertex of parabola is $(2,-1)$ and equation of its directrix is $4x-3y=21$, then the length of latus rectum 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 $PQ$ be a focal chord of the parabola ${y}^{2}=4x$ such that it subtends an angle of $\frac{\pi }{2}$ at the point $(3,0)$. Let the line segment $PQ$ be also a focal chord of the ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,{a}^{2}>{b}^{2}$. If $e$ is the eccentricity of the ellipse $E$, then the value of $\frac{1}{{e}^{2}}$ is equal to

The line $y=x+1$ meets the ellipse $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{2}=1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then ${(3r)}^{2}$ is equal to

Let $A(\frac{3}{\sqrt{a}},\sqrt{a}),a>0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. If $D(3\mathrm{cos}\theta ,a\mathrm{sin}\theta )$, is a point in the fourth quadrant such that the maximum area of $\Delta ACD$ is $12$ square units, then $a$ is equal to _____