Question

Write the angle between the curves y² = 4x and x² = 2y − 3 at the point (1, 2) ?

Answer 1

\[\text { Given }: \]

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

\[ x^2 = 2y - 3 . . . \left( 2 \right)\]

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

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

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

\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 1, 2 \right) = \frac{2}{2} = 1\]

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

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

\[ \Rightarrow \frac{dy}{dx} = x\]

\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( 1, 2 \right) = 1\]

\[\text { Thus, we get }\]

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

\[ \Rightarrow \tan \theta = \left| \frac{1 - 1}{1 + 1} \right|\]

\[ \Rightarrow \tan \theta = 0\]

\[ \Rightarrow \theta = 0^o\]