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Q.

The equations of two sides of variable triangle are $x = 0 and y = 3$ and the third side is a tangent to the parabola $y^{2} = 8 x$ The locus of the circumcentre is:

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a

$4 y^{2} + 4 x - 18 y + 18 = 0$

b

$y^{2} - 4 x + 9 y + 18 = 0$

c

$y^{2} - 4 x - 9 y + 18 = 0$

d

$y^{2} + 4 x + 9 y + 18 = 0$

answer is B.

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Detailed Solution

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Tangent is $x - yt + 2 t^{2} = 0$
Question Image
$\begin{array}{l} S (x, y) = (\frac{3 t - 2 t^{2}}{2}, \frac{2 t + 3}{2}) \\ \Rightarrow 3 t - 2 t^{2} = 2 x, \frac{2 t + 3}{2} = y \end{array}$
$\begin{array}{l} \Rightarrow 3 {(\frac{2 y - 3}{2} - 2 \frac{2 y - 3}{2})}^{2} = 2 x \\ 4 y^{2} + 4 x - 18 y + 18 = 0 \end{array}$

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