Coordinate Geometry Questions (181)

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

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

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 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

Variable pairs of chords at right angles are drawn through a point $P$ (with eccentric angle $\dfrac{x}{4}$) on the ellipse $\dfrac{x^2}{4} + y^2 = 1$ to meet the ellipse at two points, say $A$ and $B$. If the line joining $A$ and $B$ passes through a fixed point $Q = (a, b)$ and the line value of $a^2 + b^2$ can be expressed as $\dfrac{m}{n}$, where $m$ and $n$ are co-prime positive integers, submit your answer as $n - m$.

Let $(\alpha,\beta)$ be circumcenter of the triangle formed by vertices $(1,5)$, $(-3,4)$ and $(10,-31)$. Then value of $8\alpha-2\beta$ equals

Sum of length of all the common tangents of the circles $x^2+y^2-2x-8y+15=0$ and $x^2+y^2-6x-12y+43=0$ is

Consider three curves $H:(x+a)y=\lambda$, $\lambda0$; $C: x^2+y^2-21y+109=0$; $P: y^2=bx$, $b>0$. Let the line $2x-y+8=0$ touch $H$, $C$ and $P$ at $L$, $M$, $N$ respectively such that $LM=MN=\sqrt{45}$. Then $8\lambda+b+a$ is

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

A normal with slope $\frac{1}{m}$ is drawn from $P(0,-k)$ to $x^2=-12y$. The line through $Q\!\left(0,-\frac{k}{506}\right)$ parallel to the tangent at vertex meets the parabola at $R,S$. Area of $\triangle ORS = 144$ sq. units. Then $m^2$ equals

Find the largest value of $y/x$ for a point $(x,y)$ on the circle $(x-3)^2+(y-3)^2=6$.

Area bounded by the set of points $S=\{(x,y):\,||x|-1|+||y|-1|\leq1\}$ is

Chord of contact of tangents from $P,Q$ on $\dfrac{x^2}{9}+\dfrac{y^2}{16}=1$ to $\dfrac{x^2}{3}+\dfrac{y^2}{4}=1$ is normal to $2x^2+2y^2-4x-4y+1=0$. Sum of eccentric angles of $P$ and $Q$ is

A line $y = m(x-4)$ meets the $x$-axis at $P$ and the parabola $x^2 = 32y$ at $Q(x_1,y_1)$. The tangent to the parabola at $Q$ meets the $x$-axis at $R(x_2,0)$, $0 < x_2 < 6$. If the area of $\triangle PQR$ assumes a local maximum, then the value of $m$ is

TP and TQ are tangents to a parabola and $p_1$, $p_2$, $p_3$ are the lengths of perpendiculars from $P$, $T$, $Q$ respectively on any tangent to the parabola. Then $p_1$, $p_2$, $p_3$ are in

Let an ellipse have foci $S_1$ and $S_2$ with eccentricity $e = \frac{1}{2}$. For a point $P$ on the ellipse, let $\angle PS_1S_2 = \alpha$, $\angle PS_2S_1 = \beta$, $\angle S_1PS_2 = \gamma$. If $\cot\frac{\alpha}{2}$, $\cot\frac{\gamma}{2}$, $\cot\frac{\beta}{2}$ are in A.P., then $\cos(\alpha - \beta)$ is

The coordinate of point $P$ on the line $3x+2y+10=0$ such that $|PA-PB|$ is maximum, where $A=(4,2)$ and $B=(2,4)$, 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 $a+b+c=$

Circle $x^2+y^2=r^2$ meets ellipse $16x^2+25y^2=400$; $4<r<5$. Common tangent (slope $m>0$) in 2nd quadrant meets axes at $P$, $Q$. Area of $\triangle OPQ$ (O=origin) is minimum. Then $m$ is

Sum of length of all the common tangents of the circles $x^2+y^2-2x-8y+15=0$ and $x^2+y^2-6x-12y+43=0$ 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

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

A normal with slope $\frac{1}{m}$ is drawn from $P(0,-k)$ to $x^2=-12y$. The line through $Q\!\left(0,-\frac{k}{506}\right)$ parallel to the tangent at vertex meets the parabola at $R,S$. Area of $\triangle ORS = 144$ sq. units. Then $m^2$ equals

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

Find the largest value of $y/x$ for a point $(x,y)$ on the circle $(x-3)^2+(y-3)^2=6$.

Area bounded by the set of points $S=\{(x,y):\,||x|-1|+||y|-1|\leq1\}$ 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

Chord of contact of tangents from $P,Q$ on $\dfrac{x^2}{9}+\dfrac{y^2}{16}=1$ to $\dfrac{x^2}{3}+\dfrac{y^2}{4}=1$ is normal to $2x^2+2y^2-4x-4y+1=0$. Sum of eccentric angles of $P$ and $Q$ is

A line $y = m(x-4)$ meets the $x$-axis at $P$ and the parabola $x^2 = 32y$ at $Q(x_1,y_1)$. The tangent to the parabola at $Q$ meets the $x$-axis at $R(x_2,0)$, $0 < x_2 < 6$. If the area of $\triangle PQR$ assumes a local maximum, then the value of $m$ is

The radical centre of circles described on the three sides $4x-7y+10=0$, $x+y=5$, and $7x+4y=15$ of a triangle as diameters is

Let an ellipse have foci $S_1$ and $S_2$ with eccentricity $e = \frac{1}{2}$. For a point $P$ on the ellipse, let $\angle PS_1S_2 = \alpha$, $\angle PS_2S_1 = \beta$, $\angle S_1PS_2 = \gamma$. If $\cot\frac{\alpha}{2}$, $\cot\frac{\gamma}{2}$, $\cot\frac{\beta}{2}$ are in A.P., then $\cos(\alpha - \beta)$ is