The equation of the normal at the point P (2, 3) on the ellipse $9x^2+16y^2=180$is

We know that the equation of the normal at $\left(x_1,y_1\right)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is given by

$\frac{a^{2}x}{x_1}-\frac{b^{2}y}{y_1}=a^2-b^2$

The equation of the ellipse is 

$9x^2+16y^2=180\Rightarrow \frac{x^2}{20}+\frac{y^2}{{\displaystyle \frac{45}{4}}}=1$

The equation of the normal to this ellipse at (2, 3) is

$\frac{20}{2}x-\frac{45}{12}y=20-\frac{45}{4}\left[\text{ Using: }\frac{a^{2}x}{x_1}-\frac{b^{2}y}{y_1}=a^2-b^2\right]$

$\Rightarrow 10x-\frac{15}{4}y=\frac{35}{4}\Rightarrow 8x-3y=7$