Normal equations to parabola y 2-4 ax passing through point 5 a - 2 a are A- y -3 x -17 aB- x-3 y-11 aC- y - x -3 aD- y-2 x-12 a

The correct options are C y = x 3a D y = 2x + 12aNormal to y^2 = 4ax is y = tx + at^3 + 2at Since it nbsp;passes through (5a, 2a) ∴ 2a= 5at + at^3 + 2at ...

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Standard XII

Mathematics

Equation of Normal at Given Point

Normal equati...

Question

Normal equations to parabola $y^{2} = 4 a x$ passing through point $(5 a, 2 a)$ are

y = - 3 x + 17 a

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x = - 3 y + 11 a

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y = x - 3 a

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y = - 2 x + 12 a

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Solution

The correct options are
C $y = x - 3 a$
D $y = - 2 x + 12 a$
Normal to $y^{2} = 4 a x$ is
$y = - t x + a t^{3} + 2 a t$
Since it passes through $(5 a, 2 a)$
$∴ 2 a = - 5 a t + a t^{3} + 2 a t \Rightarrow 2 a = a t^{3} - 3 a t \Rightarrow t^{3} - 3 t - 2 = 0 (∵ a \neq 0)$
By observation $t = - 1$ is one root,
$\Rightarrow (t + 1) (t^{2} - t - 2) = 0 \Rightarrow (t + 1)^{2} (t - 2) = 0 ∴ t = - 1 or t = 2$
So, the equation of normal are
$t = - 1 \Rightarrow y = x - a - 2 a \Rightarrow y = x - 3 a t = 2 \Rightarrow y = - 2 x + 8 a + 4 a \Rightarrow y = - 2 x + 12 a$

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Equation of Normal at Given Point

Standard XII Mathematics

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Normal equations to parabola y2=4ax passing through point (5a,2a) are

Normal equations to parabola $y^{2} = 4 a x$ passing through point $(5 a, 2 a)$ are