Let L be a normal to the parabola y2=4x. If L passes through the point (9, 6), then L is given by [2011]

Question

Let L be a normal to the parabola $y^{2} = 4 x$ . If L passes through the point (9, 6), then L is given by [2011]

Smart Achievers · Accepted Answer

(1, 2, 4)

The equation of normal to $y^{2} = 4 x$ is $y = m x - 2 m - m^{3}$

Since the normal passes through (9, 6), $∴$ $6 = 9 m - 2 m - m^{3}$

$\Rightarrow m^{3} - 7 m + 6 = 0 \Rightarrow (m - 1) (m^{2} + m - 6) = 0$

$\Rightarrow (m - 1) (m + 3) (m - 2) = 0 \Rightarrow m = 1, 2, - 3$

$∴$ $Normals are y - x + 3 = 0 or y - 2 x + 12 = 0 or y + 3 x - 33 = 0$

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