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  • The angle made by a double ordinate of length  8 a  at the vertex of the parabola <img alt="\mathrm{ y^2=4 a x}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20y%5E2%

The angle made by a double ordinate of length  8 a  at the vertex of the parabola \mathrm{ y^2=4 a x} is 

Option: 1

\frac{\pi}{3}


Option: 2

\frac{\pi}{2}


Option: 3

\frac{\pi}{4}


Option: 4

\frac{\pi}{6}


Answers (1)

best_answer

Let P Q be a double ordinate of length 8 a.
Then \mathrm{P R=R Q=4 a}.
Coordinates of P and Q are \mathrm{(O R, 4 a)} and \mathrm{(O R,-4 a)} respectively.
Since P lies on the parabola \mathrm{y^2=4 a x}, therefore

\mathrm{(4 a)^2=4 a(O R) \Rightarrow O R=4 a}
Thus, the coordinates of P and Q are \mathrm{(4 a, 4 a)} and \mathrm{(4 a,-4 a)}
Now,
\mathrm{m_1=\text { Slope of } O P=\frac{4 a-0}{4 a-0}=1 }
and
\mathrm{ m_2=\text { Slope of } O Q=\frac{-4 a-0}{4 a-0}=-1}

Clearly, \mathrm{ m_1 m_2=-1}.
Thus, P Q makes a right angle at the vertex of the parabola.

 

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