A triangle is formed by tangents at point (2,2) on curves y^2=2 x and x^2+y^2=4 x , and line x+y+2=0

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10-2. Parabola, Ellipse, Hyperbola

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A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2 x$ and $x^2+y^2=4 x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r ^2$ is equal to $........$.

A

$10$

B

$18$

C

$15$

D

$14$

(JEE MAIN-2023)

Solution

$S_1: y^2=2 x \quad S_2: x^2+y^2=4 x$

$P (2,2)$ is common point on $S _1 \& S _2$

$T_1$ is tangent to $S_1$ at $P \quad \Rightarrow T_1: y \cdot 2=x+2$

$\Rightarrow T_1: x-2 y+2=0$

$T_2$ is tangent to $S_2$ at $P \quad \Rightarrow T_2: x \cdot 2+y \cdot 2=2(x+2)$

$\Rightarrow T_2: y=2$

$L _3: x + y +2=0$ is third line

$PQ = a =\sqrt{20}$

$QR = b =\sqrt{8}$

$RP = c =6$

$\text { Area }(\Delta PQR )=\Delta=\frac{1}{2} \times 6 \times 2=6$

$\therefore r =\frac{ abc }{4 \Delta}=\frac{\sqrt{160}}{4}=\sqrt{10} \Rightarrow r ^2=10$

Standard 11

Mathematics

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