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 __________ . [2023]

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Solution

Q.
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 __________ . [2023]

Ans.
(10)

$Given curve S_{1} : y^{2} = 2 x and S_{2} : x^{2} + y^{2} = 4 x$

$Point P (2, 2) is common on S_{1} and S_{2}$

$T_{1} is tangent to S_{1} at P$

$T_{1} : 2 y = x + 2$

$\Rightarrow T_{1} : x - 2 y + 2 = 0$

   $T_{2} is tangent to S_{2} at P$

$\Rightarrow T_{2} : 2 x + 2 y = 2 (x + 2)$

$\Rightarrow T_{2} : y = 2$

   $Line L_{3} : x + y + 2 = 0$

$Now, P Q = a = \sqrt{20}, Q R = b = \sqrt{8}, R P = c = 6$

$Area of ∆ P Q R = Δ = \frac{1}{2} \times 6 \times 2 = 6$

$∴$    $r = \frac{a b c}{4 Δ} = \frac{\sqrt{160}}{4} = \sqrt{10}, r^{2} = 10$

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