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  • The equation of a tangent to the parabola,  which makes an angle θ with the positive di

The equation of a tangent to the parabola, x^2=8 y which makes an angle θ with the positive direction of X-axis, is 

Option: 1

\begin{aligned} & y=x \tan \theta-2 \cot \theta \\ \end{aligned}


Option: 2

x=y \cot \theta+2 \tan \theta \\


Option: 3

y=x \tan \theta+2 \cot \theta \\


Option: 4

y=y \cot \theta-2 \tan \theta


Answers (1)

best_answer

Given parabola is x^2=8 y --------------(i)

Now, slope of tangent at any point ( , ) x y on the parabola (i) is 

\frac{d y}{d x}=\frac{x}{4}=\tan \theta  [ tangent is making an angle θ with the positive direction of X-axis ]

So\: \: x=4 \tan \theta\\ \Rightarrow 8 y=(4 \tan \theta)^2

From (i)

\Rightarrow y=2 \tan ^2 \theta

Now, equation of required tangent is

\begin{aligned} & \Rightarrow y-2 \tan ^2 \theta=\tan \theta(x-4 \tan \theta) \\ & \Rightarrow y=x \tan \theta-\tan ^2 \theta \\ & \Rightarrow x=y \cot \theta+2 \tan \theta \end{aligned}

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Pankaj

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