JEE Main 2022 — Coordinate Geometry

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

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 _____.

A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x-y-2=0$ at the point $B$. If the length of the line segment $AB$ is $\frac{\sqrt{29}}{3}$, then $B$ also lies on the line

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 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

Let the hyperbola $H:\frac{{x}^{2}}{{a}^{2}}-{y}^{2}=1$ and the ellipse $E:3{x}^{2}+4{y}^{2}=12$ be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$. If ${e}_{H}$ and ${e}_{E}$ are the eccentricities of $H$ and $E$ respectively, then the value of $12({e}_{H}^{2}+{e}_{E}^{2})$ is equal to _____.

Let $A(\alpha ,-2),B(\alpha ,6)$ and $C(\frac{\alpha }{4},-2)$ be vertices of a $\Delta ABC$. If $(5,\frac{\alpha }{4})$ is the circumcentre of $\Delta ABC$, then which of the following is NOT correct about $\Delta ABC$

Let the circumcentre of a triangle with vertices $A(a,3),B(b,5)$ and $C(a,b),ab>0$ be $P(1,1)$. If the line $AP$ intersects the line $BC$ at the point $Q({k}_{1},{k}_{2})$, then ${k}_{1}+{k}_{2}$ 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 a triangle be bounded by the lines ${L}_{1}:2x+5y=10$; ${L}_{2}:-4x+3y=12$ and the line ${L}_{3}$, which passes through the point $P(2,3)$, intersect ${L}_{2}$ at $A$ and ${L}_{1}$ at $B$. If the point $P$ divides the line-segment $AB$, internally in the ratio $1:3$, then the area of the triangle is equal to

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:

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

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 $P:{y}^{2}=4ax,a>0$ be a parabola with focus $S$.Let the tangents to the parabola $P$ make an angle of $\frac{\pi }{4}$ with the line $y=3x+5$ touch the parabola $P$ at $A$ and $B$. Then the value of $a$ for which $A,B$ and $S$ are collinear is:

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 $C$ be the centre of the circle ${x}^{2}+{y}^{2}-x+2y=\frac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $C$, makes an angle of $\frac{\pi }{4}$ with the line $CP$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $PQR$ (in ${\mathrm{unit}}^{2}$) is

If the circle ${x}^{2}+{y}^{2}-2gx+6y-19c=0,g,c\in \mathbb{R}$ passes through the point $(6,1)$ and its centre lies on the line $x-2cy=8$, then the length of intercept made by the circle on $x$-axis is

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 ______

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

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

A point $P$ moves so that the sum of squares of its distances from the points $(1,2)$ and $(-2,1)$ is $14$. Let $f(x,y)=0$ be the locus of $P$, which intersects the $x$-axis at the points $A,B$ and the $y$-axis at the point $C,D$. Then the area of the quadrilateral $ACBD$ 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:

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 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

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

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 _____.

The equations of the sides $AB,BC$ and $CA$ of a triangle $ABC$ are $2x+y=0,x+py=39$ and $x-y=3$ respectively and $P(2,3)$ is its circumcentre. Then which of the following is NOT true

If vertex of parabola is $(2,-1)$ and equation of its directrix is $4x-3y=21$, then the length of latus rectum is

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

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 ${P}_{1}$ 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 $x+2y=$ _____.

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 $PQ$ 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

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 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:

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

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 $H:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1,a>0,b>0$, 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 ______.

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

Let the locus of the centre $(\alpha ,\beta ),\beta >0$, of the circle which touches the circle ${x}^{2}+{(y-1)}^{2}=1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y=4$ 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 _______.

Let the lines $y+2x=\sqrt{11}+7\sqrt{7}$ and $2y+x=2\sqrt{11}+6\sqrt{7}$ be normal to a circle $C:{(x-h)}^{2}+{(y-k)}^{2}={r}^{2}$. If the line $\sqrt{11}y-3x=\frac{5\sqrt{77}}{3}+11$ is tangent to the circle $C$, then the value of ${(5h-8k)}^{2}+5{r}^{2}$ is equal to ______.

If the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ meets the line $\frac{x}{7}+\frac{y}{2\sqrt{6}}=1$ on the $x$-axis and the line $\frac{x}{7}-\frac{y}{2\sqrt{6}}=1$ on the $y$-axis, then the eccentricity of the ellipse 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

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

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?

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

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 $C$ be a circle passing through the points $A(2,-1)$ and $B(3,4)$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle ${(x-5)}^{2}+{(y-1)}^{2}=\frac{13}{2}$, then ${r}^{2}$ 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

A rectangle $R$ with end points of the one of its sides as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of $R$ is _____.

Let the foci of the ellipse $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{7}=1$ and the hyperbola $\frac{{x}^{2}}{144}-\frac{{y}^{2}}{\alpha }=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola 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

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

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

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 _____

Let the point $P(\alpha ,\beta )$ be at a unit distance from each of the two lines ${L}_{1}:3x-4y+12=0$, and ${L}_{2}:8x+6y+11=0$. If $P$ lies below ${L}_{1}$ and above ${L}_{2}$, then $100(\alpha +\beta )$ 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