Ellipses and hyperbolas

What you’ll learn

• Use eccentricity, foci, directrices and parameterisations.
• Find conic tangents and normals.

Understand the idea

For an ellipse with major axis on the x-axis, $a>b>0$ and the sum of focal distances is $2a$. Its eccentricity is between $0$ and $1$. A horizontal hyperbola has eccentricity greater than $1$, foci on the x-axis and asymptotes $y=\pm bx/a$.

The parameterisation $(a\cos t,b\sin t)$ traces an ellipse. For a hyperbola, $(a\sec t,b\tan t)$ covers both branches over suitable intervals; $(a\cosh t,b\sinh t)$ covers the right branch. Implicit or parametric differentiation gives tangents and normals. A line is tangent when its substituted intersection quadratic has a repeated root.

Key formulae

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\quad e^2=1-\frac{b^2}{a^2}$$

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,\quad e^2=1+\frac{b^2}{a^2}$$

$$\text{Foci }(\pm ae,0),\qquad\text{directrices }x=\pm a/e$$

$$\frac{xx_1}{a^2}\pm\frac{yy_1}{b^2}=1\quad\text{(tangent at }(x_1,y_1)\text{)}$$

Worked example

Find the eccentricity and foci of $x^2/25+y^2/9=1$.

1. The semi-axes are $a=5$, $b=3$.
2. Thus $e^2=1-9/25=16/25$ and $e=4/5$.
3. The focal distance from the origin is $ae=4$.

Answer: $e=4/5$; foci $(4,0)$ and $(-4,0)$.