The Ellipse and the Hyperbola: Quick Revision

A brief note on The Ellipse and the Hyperbola. Read it before a test. Try each question at the end before you open its answer.

The ellipse

• Definition: $PF_1 + PF_2 = 2a$, or $PF = e \cdot PM$ with $0 < e < 1$.
• Equation: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ with $a > b$ and $b^2 = a^2(1 - e^2)$.
• Foci and directrices: $(\pm ae, 0)$ and $x = \pm\dfrac{a}{e}$. Latus rectum: $\dfrac{2b^2}{a}$.
• Focal distances: $a \pm ex_1$. Parametric form: $(a\cos\theta, b\sin\theta)$.

The hyperbola

• Definition: $|PF_1 - PF_2| = 2a$, or $PF = e \cdot PM$ with $e > 1$.
• Equation: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ with $b^2 = a^2(e^2 - 1)$.
• Foci and directrices: $(\pm ae, 0)$ and $x = \pm\dfrac{a}{e}$. Latus rectum: $\dfrac{2b^2}{a}$.
• Asymptotes: $y = \pm\dfrac{b}{a}x$. Rectangular: $a = b$, $e = \sqrt{2}$, or $xy = c^2$.
• Parametric form: $(a\sec\theta, b\tan\theta)$. Conjugate pair: $\dfrac{1}{e^2} + \dfrac{1}{e'^2} = 1$.

Tangents and the general equation

• Tangent at $(x_1, y_1)$: $\dfrac{xx_1}{a^2} \pm \dfrac{yy_1}{b^2} = 1$. Slope form: $c^2 = a^2m^2 \pm b^2$.
• Director circle: $x^2 + y^2 = a^2 + b^2$ (ellipse), $a^2 - b^2$ (hyperbola).
• Foci and a tangent: the product of the two distances is $b^2$.
• General equation: $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2$. If $\Delta \ne 0$, then $h^2 - ab$ below, equal to or above zero gives an ellipse, a parabola or a hyperbola.

A worked example

Find the eccentricity and the foci of $\dfrac{x^2}{36} + \dfrac{y^2}{20} = 1$.

1. Axes. $a^2 = 36$ and $b^2 = 20$.
2. Focal length. $c = \sqrt{36 - 20} = 4$.
3. Answer. $e = \dfrac{4}{6} = \dfrac{2}{3}$ and the foci are $(\pm 4, 0)$.

Practice with new numbers

A program to try

The program names a conic from the numbers of its general equation and finds its eccentricity when it is an ellipse or a hyperbola with its axes along the coordinate axes.

Mistakes to avoid

• Wrong sign for $c$. Ellipse: $c^2 = a^2 - b^2$. Hyperbola: $c^2 = a^2 + b^2$.
• Wrong $a$. For an ellipse, $a$ is the larger of the two. For a hyperbola, $a^2$ is under the positive term.
• Asymptote slope. It is $\dfrac{b}{a}$.
• Forgetting $\Delta$. A pair of lines can mimic a hyperbola.

Check yourself

1. Find the eccentricity of $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$.

2. Find the asymptotes of $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$.

3. Find the tangent to $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ at $(3, 0)$.

4. What is the director circle of $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$?

5. What conic is $x^2 + xy + y^2 = 3$?

Sources

• NCERT, Mathematics for Class XI, Chapter 10 (Conic Sections), the sections on the ellipse and the hyperbola.
• OpenStax, Precalculus 2e, Chapter 12 (Analytic Geometry).