Answer
At the point $(x_0,y_0),$ the equation of a tangent line is: $~~\frac{x_0~x}{a^2}+\frac{y_0~y}{b^2} = 1$
Work Step by Step
$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$
We can find an expression for $y'$:
$\frac{2x}{a^2}+\frac{2y}{b^2}~y' = 0$
$\frac{2y}{b^2}~y' = -\frac{2x}{a^2}$
$y' = -\frac{xb^2}{ya^2}$
At the point $(x_0, y_0)$, the slope of a tangent line is $-\frac{x_0b^2}{y_0a^2}$
We can find the equation of a tangent line at the point $(x_0,y_0)$:
$y-y_0 = -\frac{x_0b^2}{y_0a^2}~(x-x_0)$
$\frac{y_0~y}{b^2}-\frac{y_0^2}{b^2} = \frac{x_0^2}{a^2}-\frac{x_0~x}{a^2}$
$\frac{x_0~x}{a^2}+\frac{y_0~y}{b^2} = \frac{x_0^2}{a^2}+\frac{y_0^2}{b^2}$
$\frac{x_0~x}{a^2}+\frac{y_0~y}{b^2} = 1$
At the point $(x_0,y_0),$ the equation of a tangent line is: $~~\frac{x_0~x}{a^2}+\frac{y_0~y}{b^2} = 1$
