JEE Main 2023 — Coordinate Geometry

Consider a circle ${C}_{1}:x{}^{2}+{y}^{2}–4x–2y=\alpha –5.$ Let its mirror image in the line $y=2x+1$ be another circle ${C}_{2}:5{x}^{2}+5{y}^{2}–10fx–10gy+36=0.$ Let $r$ be the radius of ${C}_{2}$ . Then $\alpha +r$ is equal to $________$

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

If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse ${x}^{2}+4{y}^{2}=36$ is $r$, then $12{r}^{2}$ is equal to

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

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

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 the centre of a circle $C$ be $\alpha ,\beta$ and its radius $r<8$. Let $3x+4y=24$ and $3x–4y=32$ be two tangents and $4x+3y=1$ be a normal to $C$. Then $(\alpha -\beta +r)$ 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 $A(0,1),B(1,1)$ 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

A straight line cuts off the intercepts $\mathrm{OA}=\mathrm{a}$ and $\mathrm{OB}=\mathrm{b}$ on the positive directions of $\mathrm{x}$-axis and $\mathrm{y}-$ axis respectively. If the perpendicular from origin $\mathrm{O}$ to this line makes an angle of $\frac{\pi}{6}$ with positive direction of $y$-axis and the area of $\triangle \mathrm{OAB}$ is $\frac{98}{3} \sqrt{3}$, then $\mathrm{a}^2-\mathrm{b}^2$ is equal to:

Let $A$ be the point $(1,2)$ and $B$ be any point on the curve ${x}^{2}+{y}^{2}=16$. If the centre of the locus of the point $P$, which divides the line segment $AB$ in the ratio $3:2$ is the point $C(\alpha ,\beta )$, then the length of the line segment $AC$ is

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 :

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

If the $x$-intercept of a focal chord of the parabola ${y}^{2}=8x+4y+4$ is $3$, then the length of this chord is equal to _____ .

Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along x-axis. If its minor axis subtends an angle ${60}^{\circ }$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _______.

Let $P(\frac{2\sqrt{3}}{\sqrt{7}},\frac{6}{\sqrt{7}}),Q,R$ and $S$ be four points on the ellipse $9{x}^{2}+4{y}^{2}=36$. Let $PQ$ and $RS$ be mutually perpendicular and pass through the origin. If $\frac{1}{{(PQ)}^{2}}+\frac{1}{{(RS)}^{2}}=\frac{p}{q},$ where $p$ and $q$ are coprime, then $p+q$ is equal to

The distance between the foci of the hyperbola x²/9 - y²/16 = 1 is:

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:

Two circles in the first quadrant of radii ${r}_{1}$ and ${r}_{2}$ touch the coordinate axes. Each of them cuts off an intercept of $2$ units with the line $x+y=2$. Then ${{r}_{1}}^{2}+{{r}_{2}}^{2}-{r}_{1}{r}_{2}$ 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 points of intersection of the line $ax+by=0$, $(a\neq b)$ and the circle ${x}^{2}+{y}^{2}-2x=0$ are $A(\alpha ,0)$ and $B(1,\beta )$. The image of the circle with $AB$ as a diameter in the line $x+y+2=0$ is :

If the orthocentre of the triangle, whose vertices are $(1,2),(2,3)$ and $(3,1)$ is $(\alpha ,\beta )$, then the quadratic equation whose roots are $\alpha +4\beta$ and $4\alpha +\beta$, is

The locus of the middle points of the chords of the circle ${C}_{1}:{(x-4)}^{2}+{(y-5)}^{2}=4$ which subtend an angle ${\theta }_{i}$ at the centre of the circle ${C}_{i}$, is a circle of radius ${r}_{i}$. If ${\theta }_{1}=\frac{\pi }{3}$, ${\theta }_{3}=\frac{2\pi }{3}$ and ${{r}_{1}}^{2}={{r}_{2}}^{2}+{{r}_{3}}^{2}$, then ${\theta }_{2}$ is equal to

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

The equations of the sides $AB,BC&CA$ of a triangle $ABC$ are $2x+y=0$, $x+py=21a$ $(a\neq 0)$ and $x-y=3$ respectively. Let $P(2,a)$ be the centroid of the triangle $ABC$, then ${(BC)}^{2}$ is equal to

Points $P(-3,2),Q(9,10)$ and $R(\alpha ,4)$ lie on a circle $C$ with $PR$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2x-ky=1$, then $k$ is equal to _____ .

The number of common tangents, to the circles ${x}^{2}+{y}^{2}-18x-15y+131=0$ and ${x}^{2}+{y}^{2}-6x-6y-7=0$, is

Let the eccentricity of an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ is reciprocal to that of the hyperbola $2{x}^{2}-2{y}^{2}=1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is _____.

A triangle is formed by $X$-axis, $Y$-axis and the line $3x+4y=60$. Then the number of points $P(a,b)$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiple of $a$, is _____ .

Let the point $(p,p+1)$ lie inside the region $E={(x,y):3-x\leq y\leq \sqrt{9-{x}^{2}},0\leq x\leq 3}.$ If the set of all values of $p$ is the interval $(a,b),$ then ${b}^{2}+b-{a}^{2}$ is equal to $________.$

If $(\alpha ,\beta )$ is the orthocenter of the triangle $ABC$ with vertices $A(3,–7),B(–1,2)$ and $C(4,5)$, then $9\alpha -6\beta +60$ is equal to

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 __________

A light ray emits from the origin making an angle $30^{\circ}$ with the positive $x$-axis. After getting reflected by the line $x+y=1$, if this ray intersects x-axis at Q, then the abscissa of Q 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