Parabola 2 Question 4
4. The equation of a tangent to the parabola, $x^{2}=8 y$, which makes an angle $\theta$ with the positive direction of $X$-axis, is
(2019 Main, 12 Jan, II)
(a) $y=x \tan \theta-2 \cot \theta$
(b) $x=y \cot \theta+2 \tan \theta$
(c) $y=x \tan \theta+2 \cot \theta$
(d) $x=y \cot \theta-2 \tan \theta$
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Answer:
Correct Answer: 4. (b)
Solution:
- Given parabola is $x^{2}=8 y$
Now, slope of tangent at any point $(x, y)$ on the parabola (i) is
$$ \frac{d y}{d x}=\frac{x}{4}=\tan \theta $$
$[\because$ tangent is making an angle $\theta$ with the positive direction of $X$-axis]
$$ \text { So, } x=4 \tan \theta $$
$$ \begin{aligned} & \Rightarrow \quad 8 y=(4 \tan \theta)^{2} \\ & \Rightarrow \quad y=2 \tan ^{2} \theta \quad \text { [on putting } x=4 \tan \theta \text { in Eq. (i)] } \end{aligned} $$
Now, equation of required tangent is
$y-2 \tan ^{2} \theta=\tan \theta(x-4 \tan \theta)$
$\Rightarrow y=x \tan \theta-2 \tan ^{2} \theta \Rightarrow x=y \cot \theta+2 \tan \theta$
