Parabola 2 Question 10
10. The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^{2}=4 x$ is
(2004, 1M)
(a) $\frac{\pi}{6}$
(b) $\frac{\pi}{4}$
(c) $\frac{\pi}{3}$
(d) $\frac{\pi}{2}$
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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 $y=m x+\frac{a}{m}$.
$\therefore$ Tangent to $y^{2}=4 x$ is $y=m x+\frac{1}{m}$
Since, tangent passes through $(1,4)$.
$$ \begin{aligned} & \therefore \quad 4=m+\frac{1}{m} \\ & \Rightarrow \quad m^{2}-4 m+1=0 \quad\left(\text { whose roots are } m _1 \text { and } m _2\right) \\ & \therefore \quad m _1+m _2=4 \text { and } m _1 m _2=1 \\ & \text { and } \quad\left|m _1-m _2\right|=\sqrt{\left(m _1+m _2\right)^{2}-4 m _1 m _2} \\ & =\sqrt{12}=2 \sqrt{3} \end{aligned} $$
Thus, angle between tangents
$$ \tan \theta=\left|\frac{m _2-m _1}{1+m _1 m _2}\right|=\left|\frac{2 \sqrt{3}}{1+1}\right|=\sqrt{3} \Rightarrow \theta=\frac{\pi}{3} $$
