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  •  The angle between the tangents drawn from the point (3, 4) to the parabola <img alt="\mathrm{y^2-2 y+4 x=0}" src="/latex-image/?%5Cmathrm%7By%5E2-2%20y&plu

 The angle between the tangents drawn from the point (3, 4) to the parabola \mathrm{y^2-2 y+4 x=0}  is 

 

Option: 1

\tan ^{-1}(8 \sqrt{3} / 7)


Option: 2

\tan ^{-1}(8 \sqrt{5} / 7)


Option: 3

\tan ^{-1}(3 \sqrt{5} / 7)


Option: 4

\tan ^{-1}(\sqrt{5} / 7)


Answers (1)

best_answer

The equation of the parabola is 

(y-1)^2=-4\left(x-\frac{1}{4}\right)

The equation of any tangent to this parabola is \mathrm{y-1=m\left(x-\frac{1}{4}\right)-\frac{1}{m}}

If it passes through (3, 4), then \mathrm{3=\frac{11 \mathrm{~m}}{4}-\frac{1}{\mathrm{~m}}}

\mathrm{\Rightarrow 11 \mathrm{~m}^2-12 \mathrm{~m}-4=0}

Let m1 & m2 be the roots of this equation. Then, 

\mathrm{m}_1+\mathrm{m}_2=\frac{12}{11} \text { and } \mathrm{m}_1 \mathrm{~m}_2=-\frac{4}{11}

Let θ be the angle between the tangents. Then,

\mathrm{\tan \theta=\frac{m_1-m_2}{1+m_1 m_2}=\frac{\sqrt{\left(m_1+m_2\right)^2-4 m_1 m_2}}{1+m_1 m_2} \Rightarrow \theta=\tan ^{-1}\left(\frac{8 \sqrt{5}}{7}\right)}

 

 

 

Posted by

Kuldeep Maurya

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