A circle circumscribing the triangle formed by three co-normal points passes through the vertex of the parabola $y^{2} = 4 a x$ where $(h, k)$ is the point from where three concurrent normals are drawn. The circle has the equation

2 (x^{2} + y^{2}) - (h + 2 a) + k y = 0

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2 (x^{2} + y^{2}) - 2 (h + 2 a) + k y = 0

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2 (x^{2} + y^{2}) - 2 (h + a) + k y = 0

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none of these

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Solution

The correct option is B $2 (x^{2} + y^{2}) - 2 (h + 2 a) + k y = 0$
The equation of the normal at $(a m^{2}, 2 a m)$ is
$y + m x = 2 a m + a m^{3}$
It passes through $(h, k)$ .
$a m^{3} + m (2 a - h) - k = 0$
The roots are $m_{1}, m_{2}$ and $m_{3}$
$\sum m_{1} = 0, \sum m_{1} m_{2} = \frac{2 a - h}{a}$
$m_{1} m_{2} m_{3} = \frac{k}{a}$
Let equation of circle be $x^{2} + y^{2} + 2 g x + 2 f y + C = 0$
It passes through $(a m^{2}, 2 a m)$
$a^{2} m^{4} + 4 a^{2} m^{2} + 2 a g m^{2} + 4 a f m + C = 0$
$a^{2} m^{4} + m^{2} (4 a^{2} + 2 a g) + 4 a f m + C = 0$
Its roots are $m_{1}, m_{2}, m_{3}$ and $m_{4}$
$m_{1} + m_{2} + m_{3} + m_{4} = 0$
$∵ m_{1} + m_{2} + m_{3} = 0$
$\Rightarrow m_{4} = 0 \Rightarrow$ the circle passes $(0, 0)$
$m_{1} m_{2} + m_{2} m_{3} + m_{3} m_{4} + m_{4} m_{1} + m_{1} m_{3} + m_{2} m_{4}$
$= \frac{4 a^{2} + 2 a g}{a^{2}}$
$\Rightarrow \frac{2 a - h}{a} = \frac{4 a^{2} + 2 a g}{a^{2}}$
$\Rightarrow 2 a - h = 4 a + 2 g$
$\Rightarrow g = \frac{- h - 2 a}{2}$
$m_{1} m_{2} m_{3} + m_{2} m_{3} m_{4} + m_{3} m_{4} m_{1} + m_{4} m_{1} m_{2} = \frac{- 4 a f}{a^{2}}$
$\Rightarrow \frac{k}{a} = \frac{- 4 a f}{a^{2}}$
$\Rightarrow f = \frac{- k}{4}$
The equation of circle is,
$x^{2} + y^{2} - (h + 2 a) x + \frac{k}{2} y = 0$
$\Rightarrow 2 (x^{2} + y^{2}) - 2 (h + 2 a) + k y = 0$

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A circle circumscribing the triangle formed by three co-normal points passes through the vertex of the parabola y2=4ax where (h,k) is the point from where three concurrent normals are drawn. The circle has the equation

A circle circumscribing the triangle formed by three co-normal points passes through the vertex of the parabola $y^{2} = 4 a x$ where $(h, k)$ is the point from where three concurrent normals are drawn. The circle has the equation