SATHEE: Parabola 2 Question 4

Parabola 2 Question 4

4. 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

(2019 Main, 12 Jan, II)

(a) $y = x \tan θ - 2 \cot θ$

(b) $x = y \cot θ + 2 \tan θ$

(d) $x = y \cot θ - 2 \tan θ$

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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

$\frac{d y}{d x} = \frac{x}{4} = \tan θ$

$[∵$ tangent is making an angle $θ$ with the positive direction of $X$ -axis]

$So, x = 4 \tan θ$

$\begin{aligned} \Rightarrow 8 y = (4 \tan θ)^{2} \\ \Rightarrow y = 2 \tan^{2} θ [on putting x = 4 \tan θ in Eq. (i)] \end{aligned}$

Now, equation of required tangent is

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

$\Rightarrow y = x \tan θ - 2 \tan^{2} θ \Rightarrow x = y \cot θ + 2 \tan θ$

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Parabola 2 Question 4

4. The equation of a tangent to the parabola, x2=8y, which makes an angle θ with the positive direction of X-axis, is

Answer:

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4. 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