A normal chord of the parabola $y^{2} = 4 x$ makes an angle $45^{0}$ with the axis of the parabola. Then its length is

8 \sqrt{2}

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10 \sqrt{2}

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6 \sqrt{3}

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4 \sqrt{3}

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Solution

The correct option is A $8 \sqrt{2}$
Eq. of normal to parabola is.
$y + t x = 2 a t + a t^{3}$
$y^{'} = - t = 1$
$t = - 1$
$y - x = - 2 - 1$
$y - x + 3 = 0$
$y^{2} = 4 (y + 3)$
$y^{2} - 4 y - 12 = 0$
$x^{2} - 6 x + 9 = 4 x$
$x^{2} - 10 x + 9 = 0$
$\sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}} =$ length
$\sqrt{(x_{1} + x_{2})^{2} - 4 x_{1} x_{2} + (y_{1} + y_{2})^{2} - 4 y_{1} y_{2}} = l$
$\sqrt{10^{2} - (4 \times 9) + (4)^{2} + 4 \times 12} = l$
$l = \sqrt{128}$
$l = 8 \sqrt{2}$

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