The equation of a tangent to the parabola,

Question:

The equation of a tangent to the parabola, $\mathrm{x}^{2}=8 \mathrm{y}$, which makes an angle $\theta$ with the positive direction of $\mathrm{x}$-axis, is :

  1. $x=y \cot \theta+2 \tan \theta$

  2. $x=y \cot \theta-2 \tan \theta$

  3. $y=x \tan \theta-2 \cot \theta$

  4. $y=x \tan \theta+2 \cot \theta$


Correct Option: 1

Solution:

$x^{2}=8 y$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}}{4}=\tan \theta$

$\therefore \quad \mathrm{x}_{1}=4 \tan \theta$

$\mathrm{y}_{1}=2 \tan ^{2} \theta$

Equation of tangent :-

$y-2 \tan ^{2} \theta=\tan \theta(x-4 \tan \theta)$

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

 

 

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