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Parabola

Section: Mathematics

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Question : 6 of 15

Marks: +1, -0

The angle between the tangents drawn from the point

(1,4)

to the parabola

y^2=4 x

[AP EAPCET 21 May 2025 S2]

$\;\frac{\pi}{6}$
$\;\frac{\pi}{4}$
$\;\frac{\pi}{3}$
$\;\frac{\pi}{2}$

Solution:

y^2=4 x

Equation of tangent to the parabola

y^2=4 a x \; \text{ " is " } \; y=m x+\frac{a}{m}

\because

Equation of tangent is

y=m x+\frac{1}{m}

Given, tangents passes through

(1,4)

4=m+\frac{1}{m}

\Rightarrow \;\; m^2-4 m+1=0

\Rightarrow \;\; m_1+m_2=4, \;\; m_1 m_2=1

\because \tan \alpha \;=\;\frac{m_1-m_2}{1+m_1 m_2}

=\;\;\frac{\sqrt{ (m_1+m_2)^2-4 m_1 m_2 }}{1+m_1 m_2}

=\;\;\frac{\sqrt{16-4}}{1+1}=\;\frac{\sqrt{12}}{2}=\;\frac{2\sqrt{3}}{2}=\sqrt{3}

\therefore \;\; \alpha = \tan^{-1} \sqrt{3} = \frac{\pi}{3}

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