Question

Show that the following curve intersect orthogonally at the indicated point y² = 8x and 2x² +  y² = 10 at  \[\left( 1, 2\sqrt{2} \right)\] ?

Answer 1

\[y^2 = 8x . . . \left( 1 \right)\]

\[2 x^2 + y^2 = 10 . . . \left( 2 \right)\]

\[\text { Given point is }\left( 1, 2\sqrt{2} \right)\]

\[\text { Differentiating (1) w.r.t.x,}\]

\[2y\frac{dy}{dx} = 8\]

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

\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( 1, 2\sqrt{2} \right) = \frac{4}{2\sqrt{2}} = \sqrt{2}\]

\[\text { Differentiating (2) w.r.t.x,}\]

\[4x + 2y\frac{dy}{dx} = 0\]

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

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

\[\text { Since,} m_1 \times m_2 = - 1\]

Hence,  the given curves intersect orthogonally at the given point.