Tangent to an Ellipse

Definition (as for the parabola). A tangent to an ellipse is a straight line that intersects the curve in two coincident points — equivalently, the limiting position of a secant as its two points of intersection merge into one.

Finding the tangent, using calculus. We want the tangent to $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ at a point $P(x_1,y_1)$ on it.

Differentiate implicitly with respect to $x$:

At $P(x_1,y_1)$, the slope is $-\dfrac{b^2}{a^2}\cdot\dfrac{x_1}{y_1}$. By point-slope form:

Dividing throughout by $a^2b^2$:

But $P(x_1,y_1)$ lies on the ellipse, so the right-hand side equals $1$. Hence the point-form tangent is

— the same "replace $x^2\to xx_1,\,y^2\to yy_1$" pattern seen for the parabola.

Eccentric-angle form. Since $P(x_1,y_1)=(a\cos\theta_1,\,b\sin\theta_1)$ for its eccentric angle $\theta_1$, substituting into the point-form tangent: …