Question

The angle of intersection of the parabolas y² = 4 ax and x² = 4ay at the origin is ____________ .

Options

• π/6

• π/3

• π/2

• π/4

Answer 1

π/2

 

\[\text { Given }: \]

\[ y^2 = 4ax . . . \left( 1 \right)\]

\[ x^2 = 4ay . . . \left( 2 \right)\]

\[\text { Point } =\left( 0, 0 \right)\]

\[\text { On differentiating (1) w.r.t.x,we get }\]

\[2y \frac{dy}{dx} = 4a\]

\[ \Rightarrow \frac{dy}{dx} = \frac{2a}{y}\]

\[ \Rightarrow m_1 = \infty \]

\[\text { Now, on differentiating (2) w.r.t.x, we get }\]

\[2x = 4a\frac{dy}{dx}\]

\[ \Rightarrow \frac{dy}{dx} = \frac{x}{2a} = 0\]

\[ \therefore \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = \left| \frac{\infty}{1 + 0} \right| = \infty \]

\[ \Rightarrow \theta = \tan^{- 1} \infty = \frac{\pi}{2}\]