SATHEE: Parabola 2 Question 31

Parabola 2 Question 31

10. The angle between the tangents drawn from the point $(1, 4)$ to the parabola $y^{2} = 4 x$ is

(2004, 1M)

(a) $\frac{π}{6}$

(b) $\frac{π}{4}$

(d) $\frac{π}{2}$

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Answer:

Correct Answer: 10. $\sqrt{c - \frac{1}{4}}, \frac{1}{2} \leq c \leq 5$

Solution:

We know, tangent to $y^{2} = 4 a x$ is $y = m x + \frac{a}{m}$ .

$∴$ Tangent to $y^{2} = 4 x$ is $y = m x + \frac{1}{m}$

Since, tangent passes through $(1, 4)$ .

$\begin{aligned} ∴ 4 = m + \frac{1}{m} \\ \Rightarrow m^{2} - 4 m + 1 = 0 (whose roots are m_{1} and m_{2}) \\ ∴ m_{1} + m_{2} = 4 and m_{1} m_{2} = 1 \\ and | m_{1} - m_{2} | = \sqrt{{(m_{1} + m_{2})}^{2} - 4 m_{1} m_{2}} \\ = \sqrt{12} = 2 \sqrt{3} \end{aligned}$

Thus, angle between tangents

$\tan θ = | \frac{m_{2} - m_{1}}{1 + m_{1} m_{2}} | = | \frac{2 \sqrt{3}}{1 + 1} | = \sqrt{3} \Rightarrow θ = \frac{π}{3}$

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Parabola 2 Question 31

10. The angle between the tangents drawn from the point (1,4) to the parabola y2=4x is

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10. The angle between the tangents drawn from the point $(1, 4)$ to the parabola $y^{2} = 4 x$ is