Coordinate Geometry Questions - JEE

In a parabola $y^2=4ax$, two points $P$ and $Q$ are taken such that the tangents drawn to the parabola at these points meet at the directrix at $R$. The focus (locus of circumcenter of $\triangle PQR$) will be

The maximum value of $\left|\sqrt{(x^2-2)^2+(x-3)^2} - \sqrt{(x^2+2)^2+x^2}\right|$ is

Let points $S_1$ and $S_2$ (lying on positive $x$-axis) be the foci of $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ and $\dfrac{x^2}{a^2}-\dfrac{y^2}{c^2}=\dfrac{1}{4}$. If a point of intersection is equidistant from $S_1$ and $S_2$ and $a:b=\sqrt{7}:\sqrt{6}$, then $a:c$ is

In $\triangle ABC$, $X$ and $Y$ are the feet of perpendiculars drawn from $A$ to the internal angle bisectors of $B$ and $C$. The slopes of lines making $45°$ with line $XY$ are (Given slope of $BC = 2$)

Three distinct chords of $x^2+4y^2=2000$ from $P(0,a)$ are bisected by $x^2=20y$. Exhaustive set of $a$ is $(k_1,k_2)$. Number of positive integral solutions of $x+y=k_2-k_1$ is

On the hyperbola $y^2 - x^2 = 1$, consider a point $P$ with abscissa $n$ (integer). Let $d_n$ be the shortest distance from $P$ to the line $y = x$. Then $\lim_{n \to \infty} n \cdot d_n$ equals

Circles $C_1:x^2+y^2=625$, $C_2:(x-a)^2+y^2=576$, $a\in(1,49)$. Point $P$ is on both circles with $\angle QPR=\cos^{-1}\!\left(\dfrac{\sqrt{481}}{25}\right)$ ($Q,R$ are centers). Length of common tangent $=\sqrt{1295}$. Common chord length is

Circle $x^2+y^2=r^2$ meets ellipse $16x^2+25y^2=400$; $40$) in 2nd quadrant meets axes at $P$, $Q$. Area of $\triangle OPQ$ (O=origin) is minimum. Then $m$ is

If a circle is inscribed in ellipse $x^2+4y^2=4$, then the range of radius of the circle is

A line $L: 2x-y+5=0$ is tangent to the hyperbola $H\equiv\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that the foot of perpendicular from the foci of $H$ on $L$ is $\left(\frac{\sqrt{3}-2\sqrt{5}}{\sqrt{5}},\frac{2\sqrt{3}+\sqrt{5}}{\sqrt{5}}\right)$. If $H$ intersects an ellipse $E\equiv\frac{x^2}{25}+\frac{y^2}{\lambda^2}=1$ orthogonally, then eccentricity of $E$ is

Circle $C$ touches $x=2y$ at $(2,1)$ and intersects $C_1:x^2+y^2+2y-5=0$ at two points $P,Q$ such that $PQ$ is a diameter of $C_1$. Length of diameter of $C$ is

The eccentricity of ellipse $3x^2+4y^2=12$ is changed at the rate of $0.1$/sec. The time in seconds such that the ellipse becomes an auxiliary circle is

The number of integral values of $k$ for which the line $3x+4y-k=0$ lies between the circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2-18x-12y+113=0$, without cutting a chord on either circle, is equal to

Given lines $\dfrac{x}{a}+\dfrac{y}{b}=1$ and $ax+by=1$ are two variable lines, $a$ and $b$ being parameters connected by $a^2+b^2=ab$. The locus of the point of intersection is

If the circles $x^2+y^2+10\alpha x+\beta y+\alpha=0$ and $x^2+y^2-5\alpha x+\gamma y-1=0$ intersect in two distinct points $A$ and $B$, then the line $15x+\delta y-\alpha=0$ passes through $A$ and $B$ for

Area bounded by the straight lines $x^2y-y^3-x^2+5y^2-8y+4=0$ (in sq. units) is

In a parabola $y^2=4ax$, two points $P$ and $Q$ are taken such that the tangents drawn to the parabola at these points meet at the directrix at $R$. The focus (locus of circumcenter of $\triangle PQR$) will be

Let $L_1: \frac{x-1}{1} = \frac{y}{-1} = \frac{z-1}{3}$ and $L_2: \frac{x-1}{-3} = \frac{y}{-1} = \frac{z-1}{1}$ be two lines. Let $L: \frac{x-\alpha}{l} = \frac{y-1}{m} = \frac{z-\gamma}{-2}$ be a line that lies in the plane containing $L_1$ and $L_2$, passes through their point of intersection, and bisects the acute angle between them. Then which of the following statements is/are TRUE?

Consider two points $A(x_1,y_1)$ and $B(x_2,y_2)$ on the graph of $y=\dfrac{1}{x}$ such that $0<x_1<x_2$ and $OA\perp OB$ (where $O$ is the origin). Let $C$ be the midpoint of segment $AB$. Then:

Circle $C$ touches $x=2y$ at $(2,1)$ and intersects $C_1:x^2+y^2+2y-5=0$ at two points $P,Q$ such that $PQ$ is a diameter of $C_1$. Length of diameter of $C$ is

If pair of variable straight lines $x^2+4y^2+\alpha xy=0$ ($\alpha$ real) cuts the ellipse $x^2+4y^2=4$ at $A$ and $B$, then the locus of point of intersection of tangents at $A$ and $B$ of ellipse is

Let $H_n: \dfrac{x^2}{1+n}-\dfrac{y^2}{3+n}=1$, $n\in\mathbb{N}$. Let $k$ be the smallest even value of $n$ such that eccentricity of $H_k$ is rational. If $\ell$ is the length of the latus rectum of $H_k$, then $21\ell$ is equal to

Two sides of a triangle have combined equation $x^2-3y^2-2xy+8y-4=0$. The third side passes through $(-5,1)$ and the origin is interior. If the range of slope of the third line is $(a,b)$, then $\dfrac{a+\frac{1}{b^2}}{4}$ is

Two circles $x^2+y^2+14x-6y+40=0$ and $x^2+y^2-2x+6y+7=0$ have centres $C_1$, $C_2$. Another circle with centre $C_3$ on line $3x+4y-16=0$ touches $C_1$ externally and minimises $C_1C_2+C_2C_3+C_3C_1$. Its equation $x^2+y^2+ax+by+c=0$ gives $a+b+c=$

If area bounded by an ellipse with its auxiliary circle equals area bounded by its auxiliary circle and director circle, then eccentricity of ellipse is

Let $P$ be any point on $x-y+3=0$ and $A=(3,4)$. If the family of lines $(3\sec\theta+5\csc\theta)x+(7\sec\theta-3\csc\theta)y+11(\sec\theta-\csc\theta)=0$ are concurrent at $B$ for all permissible $\theta$, then max value of $|PA-PB|$ is

Let $H_L$ and $H_R$ be two branches of a hyperbola with $A,B$ as endpoints of latus rectum on $H_L$. If $C$ is on $H_R$ such that $\angle ACB$ is never obtuse, then maximum possible eccentricity is

Let $S=\{(x,y)\in\mathbb{R}\times\mathbb{R}: y^2\leq8x,\;y^2\geq32-8x,\;x+y-6\geq0,\;4x-3y-8\leq0,\;x\geq0,\;y\geq0\}$. If area $=A$, value of $21A+792$ is

A rectangular hyperbola and ellipse have coincident endpoints of latus rectum. Eccentricity of ellipse equals

Consider two circles $C_1:(x-1)^2+(y-4)^2=16$ and $C_2:(x-13)^2+(y-9)^2=81$. If a circle of radius $r$ touches the $x$-axis and both $C_1$ and $C_2$ externally, then $r$ is equal to

Three circles of radii $a,b,c$ ($a<b<c$) touch each other externally. If they have $x$-axis as a common tangent, then

Three distinct chords of $x^2+4y^2=2000$ from $P(0,a)$ are bisected by $x^2=20y$. Exhaustive set of $a$ is $(k_1,k_2)$. Number of positive integral solutions of $x+y=k_2-k_1$ is

Let $A$, $B$, $C$ be 3 points on the parabola $y^2=4x$. Let $D$, $E$, $F$ be midpoints of $AB$, $BC$ and $AC$ respectively. If $G$ is the centroid of $\triangle DEF$ and normals to the parabola at $A$, $B$ and $C$ are concurrent at $H(5,1)$, then slope of the line $GH$ is

If a circle is inscribed in ellipse $x^2+4y^2=4$, then the range of radius of the circle is

If the pair of perpendicular lines $4x^2 + by^2 + 2\cos\theta\cdot xy + 12x + 2\sin^2\theta\cdot y + c = 0$, $\theta \in \left(\frac{3\pi}{2}, 2\pi\right)$ intersect on the $x$-axis, then the area of the triangle formed by the given pair of lines and $y = 2$ is