Equation of the Tangent to the Ellipse

When does a line just touch the ellipse?

Take a line $y = mx+c$ and an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$. Substituting the line into the ellipse to find their intersection points leads to a quadratic in $x$:

A line meets a conic in general at two points; it is a tangent exactly when those two points coincide, i.e. when this quadratic has a repeated root — when its discriminant is zero. Working out $\text{(discriminant)}=0$ for the equation above and simplifying gives a strikingly clean condition:

So for every slope $m$, there are exactly two tangents with that slope, $y = mx \pm \sqrt{a^2m^2+b^2}$ — one on each side of the ellipse, which makes sense by symmetry.

Tangent at a known point on the ellipse

A more useful form in practice is the tangent at a specific point $P(x_1,y_1)$ that is already known to lie on the ellipse. Writing $S \equiv \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-1$, define the shorthand $S_1 \equiv \dfrac{xx_1}{a^2}+\dfrac{yy_1}{b^2}-1$ (replace one $x$ and one $y$ in $S$ by the coordinates of $P$). The chord joining two points $(x_1,y_1)$ and $(x_2,y_2)$ on the ellipse turns out to have the simple equation $S_1+S_2=S_{12}$; letting the second point slide along the curve until it merges into the first (the chord becoming a tangent) collapses this to:

This is easy to remember: take the ellipse equation and replace $x^2$ by $x\cdot x_1$, and $y^2$ by $y\cdot y_1$.

Tangent in terms of the eccentric angle

Substituting the parametric point $P(\theta) = (a\cos\theta, b\sin\theta)$ for $(x_1,y_1)$ in $S_1=0$ gives the tangent at the point $\theta$:

This form is often quicker to use when a problem is phrased in terms of the eccentric angle rather than raw coordinates.

Worked Example

Problem. (a) Find the equation of the tangent to the ellipse $\dfrac{x^2}{20}+\dfrac{y^2}{5}=1$ at the point $(4,1)$. (b) Determine whether the line $y = x + 5$ is a tangent to this ellipse.

Solution.

(a) Here $a^2=20$, $b^2=5$, and $(x_1,y_1)=(4,1)$ — check it lies on the ellipse: $16/20+1/5 = 0.8+0.2=1$. ✓ The tangent is $S_1=0$: …