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  • The angle between the tangents drawn from the point  to the parabola <img alt="y^2=4 x" src=

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

Option: 1

\begin{aligned} & \frac{\pi}{6} \\ \end{aligned}


Option: 2

\frac{\pi}{4} \\


Option: 3

\frac{\pi}{3} \\


Option: 4

\frac{\pi}{2}


Answers (1)

best_answer

Tangent to    y^2=4 a x \text { is } y=m x+\frac{a}{m}

\thereforeTangent to   y^2=4 x \text { is } y=m x+\frac{1}{m}

Since, tangent passes through (1,4)

\begin{aligned} & \therefore 4=m+\frac{1}{m} \\ & \Rightarrow m^2-4 m+1=0 \\ & \therefore m_1+m_2=4 \text { and } m_1 m_2=1 \end{aligned}   and 

\begin{aligned} & \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

\begin{aligned} & =\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} \end{aligned}

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