For the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, which of the following is the equation of the tangent at the point $(x_1,y_1)$ on the ellipse?

For the conic $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, replacing $x^2$ by $xx_1$ and $y^2$ by $yy_1$ gives the tangent at $(x_1,y_1)$. So the tangent equation is $\frac{xx_1}{a^2}+\frac{yy_1}{b^2}=1$. The other options do not match the standard tangent form.