Length of normal chord of parabola y2 - 4x which makes an inclination of -4 with the positive direction of x-axis is

Normal of y^2 = 4ax at (at^2,2at) is nbsp; y = tx + at^3 + 2at ⇒ slope nbsp; = t = tanπ4 ⇒ t = 1 Normal at point P(t = 1) meets y^2 = 4x again at Q(t_0) ...

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Standard XII

Mathematics

Equation of Normal at Given Point

Length of nor...

Question

Length of normal chord of parabola $y^{2} = 4 x$ which makes an inclination of $\frac{π}{4}$ with the positive direction of $x -$ axis is

8

units

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8 \sqrt{2}

units

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4

units

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4 \sqrt{2}

units

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Solution

The correct option is B $8 \sqrt{2}$ units
Normal of $y^{2} = 4 a x$ at $(a t^{2}, 2 a t)$ is
$y = - t x + a t^{3} + 2 a t$
$\Rightarrow$ slope $= - t = tan \frac{π}{4}$
$\Rightarrow t = - 1$
Normal at point $P (t = - 1)$ meets $y^{2} = 4 x$ again at $Q (t_{0})$
$\Rightarrow t_{0} = - t - \frac{2}{t} = 1 + 2 = 3$
$∴ Q = (a t_{0}^{2}, 2 a t_{0}) = (9, 6) [∵ a = 1]$
Also, $P = (1, - 2)$
$∴$ Length of normal chord $= \sqrt{(9 - 1)^{2} + (6 - (- 2))^{2}} = 8 \sqrt{2}$ units

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Length of normal chord of parabola y2=4x which makes an inclination of π4 with the positive direction of x−axis is

Length of normal chord of parabola $y^{2} = 4 x$ which makes an inclination of $\frac{π}{4}$ with the positive direction of $x -$ axis is