Answer
$\frac{x_{0}x}{a^{2}}-\frac{y_{0}y}{b^{2}}$ = $\frac{x_{0}^{2}}{a^{2}}-\frac{y_{0}^{2}}{b^{2}}$ = 1
Work Step by Step
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}$ = $1$
Differentiating
$\frac{2x}{a^{2}}-\frac{2yy'}{b^{2}}$ = $0$
$y'$ = $\frac{b^{2}x}{a^{2}y}$
an equation of the tangent line at $(x_{0},y_{0})$ is
$y-y_{0}$ = $\frac{b^{2}x}{a^{2}y}(x-x_{0})$
Multiply both side by $\frac{y_{0}}{b^{2}}$
$\frac{y_{0}y}{b^{2}}-\frac{y_{0}^{2}}{b^{2}}$ = $\frac{x_{0}x}{a^{2}}-\frac{x_{0}^{2}}{a^{2}}$
since $(x_{0},y_{0})$ lies on the hyperbola
$\frac{x_{0}x}{a^{2}}-\frac{y_{0}y}{b^{2}}$ = $\frac{x_{0}^{2}}{a^{2}}-\frac{y_{0}^{2}}{b^{2}}$ = 1
