Question

Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is  parallel to y-axis ?

Answer 1

The slope of the y-axis is \[\infty\].

Let (x₁, y₁) be the required point.

Given:

\[\text { Since, the point lies on the curve } . \]

\[\text { Hence, }\frac{{x_1}^2}{9} + \frac{{y_1}^2}{16} = 1 . . . \left( 1 \right)\]

\[\frac{x^2}{9} + \frac{y^2}{16} = 1 \]

\[ \Rightarrow \frac{2x}{9} + \frac{2y}{16}\frac{dy}{dx} = 0\]

\[ \Rightarrow \frac{y}{16}\frac{dy}{dx} = \frac{- x}{9}\]

\[ \Rightarrow \frac{dy}{dx} = \frac{- 16x}{9y}\]

\[\text { Now,} \]

\[\text { Slope of the tangent at }\left( x, y \right)= \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =\frac{- 16 x_1}{9 y_1}\]

\[\text { Slope of the tangent at }\left( x_1 , y_1 \right)= \text { Slope of they-axis [Given }]\]

\[ \therefore \frac{- 16 x_1}{9 y_1} = \infty \]

\[ \Rightarrow \frac{9 y_1}{- 16 x_1} = 0\]

\[ \Rightarrow y_1 = 0\]

\[ \Rightarrow \frac{{x_1}^2}{9} + 0 = 1 [\text { From eq.} (1)]\]

\[ \Rightarrow {x_1}^2 = 9\]

\[ \Rightarrow x_1 = \pm 3\]

\[\text { Thus, the required points are }\left( 3, 0 \right)\text { and }\left( - 3, 0 \right).\]