Ellipse

An ellipse is the set of points $P$ such that the sum of distances from $P$ to two fixed points (the foci) is constant. This sum equals the length of the major axis, $2a$ .

A garden trick to draw one: pin the ends of a loose string at two points and trace with a pencil keeping the string taut. The resulting curve is an ellipse with the pins at the foci.

Standard form (horizontal major axis)

Place the foci at $F_1(-c, 0)$ and $F_2(c, 0)$ with $c > 0$ , and let the constant sum of distances be $2 a$ (so $a > c$ ). Setting $b^2 = a^2 - c^2$ : $\boxed{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b > 0.}$

Key features.

Centre: origin.
Vertices on major axis: $(\pm a, 0)$ .
Vertices on minor axis: $(0, \pm b)$ .
Foci: $(\pm c, 0)$ , where $c = \sqrt{a^2 - b^2}$ .
Eccentricity: $e = c/a$ , $0 < e < 1$ .
Directrices: $x = \pm a/e$ .
Latus rectum: length $\dfrac{2 b^2}{a}$ , perpendicular chords through foci.
Major axis length: $2 a$ .
Minor axis length: $2 b$ .

Standard form (vertical major axis)

If $a < b$ , the major axis is along the $y$ -axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad b > a > 0.$ Now foci are at $(0, \pm c)$ with $c^2 = b^2 - a^2$ , vertices on major axis at $(0, \pm b)$ , eccentricity $e = c/b$ .

Some books label the larger semi-axis $a$ always; we have used $a$ for the $x$ -semi-axis throughout for clarity. Always check which is larger.

Why the focal-sum is $2 a$

From the definition, $|PF_1| + |PF_2| = 2 a$ for every $P$ on the ellipse. At the vertex $(a, 0)$ : $|PF_1| = a + c$ , $|PF_2| = a - c$ , sum $= 2 a$ . ✓

Worked examples

Example 1. Find centre, foci, vertices, eccentricity, and latus rectum of $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ .

$a^2 = 25, b^2 = 9 \Rightarrow a = 5, b = 3$ . $c^2 = 16 \Rightarrow c = 4$ . Centre: origin. Vertices on major: $(\pm 5, 0)$ ; on minor: $(0, \pm 3)$ . Foci: $(\pm 4, 0)$ . $e = 4/5$ . Latus rectum: $2 b^2 / a = 18/5$ .

Example 2. Find centre, foci, vertices of $\dfrac{x^2}{4} + \dfrac{y^2}{25} = 1$ .

$a^2 = 4, b^2 = 25$ so $b > a$ , vertical major axis. $c^2 = b^2 - a^2 = 21 \Rightarrow c = \sqrt{21}$ . Foci: $(0, \pm\sqrt{21})$ . Major vertices: $(0, \pm 5)$ . Minor vertices: $(\pm 2, 0)$ . $e = \sqrt{21}/5$ .

Example 3. Find the equation of the ellipse with foci $(\pm 3, 0)$ and major-axis length $10$ .

$2 a = 10 \Rightarrow a = 5$ . $c = 3$ . $b^2 = a^2 - c^2 = 16$ . Equation: $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$ .

Example 4. Find the equation of the ellipse with vertices $(\pm 6, 0)$ and eccentricity $1/3$ .

$a = 6, e = 1/3$ . $c = ea = 2$ . $b^2 = a^2 - c^2 = 36 - 4 = 32$ . Equation: $\dfrac{x^2}{36} + \dfrac{y^2}{32} = 1$ .

Example 5. Find the focal distances of the point $(3, 4/5)$ on the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ .

Verify on ellipse: $9/25 + (16/25)/9 = 9/25 + 16/225 = 81/225 + 16/225 = 97/225 \ne 1$ . The point is not on this ellipse. Let me reconsider: the point on the ellipse is $(3, 12/5)$ . Check: $9/25 + (144/25)/9 = 9/25 + 16/25 = 1$ ✓.

For $(3, 12/5)$ : $|PF_1| = a + e x = 5 + (4/5)(3) = 5 + 12/5 = 37/5$ . $|PF_2| = a - e x = 5 - 12/5 = 13/5$ . Sum: $50/5 = 10 = 2a$ . ✓

(Useful formula: for any point $(x, y)$ on $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ with foci on $x$ -axis, $|PF_1| = a + e x$ , $|PF_2| = a - e x$ .)

Try it yourself

Find centre, foci, vertices of $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$ .
Find centre, foci, vertices of $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$ .
Find eccentricity of $\dfrac{x^2}{49} + \dfrac{y^2}{36} = 1$ .
Equation of ellipse with foci $(\pm 4, 0)$ and vertices $(\pm 5, 0)$ .
Equation of ellipse with foci $(0, \pm 3)$ and major-axis length $10$ .
Equation of ellipse with eccentricity $1/2$ and vertices $(\pm 4, 0)$ .
Length of latus rectum of $\dfrac{x^2}{36} + \dfrac{y^2}{20} = 1$ .
Find the points on the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ whose distance from $(4, 0)$ equals $\dfrac{14}{5}$ .
Equation of the ellipse with centre at origin, major axis along $y$ -axis, passing through $(3, 2)$ and $(1, 6)$ .
Find the eccentricity, foci, and length of latus rectum of $9 x^2 + 25 y^2 = 225$ .
A point on an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ has focal distances $3$ and $5$ . Find $a$ .
The eccentricity of an ellipse is $\dfrac{\sqrt{3}}{2}$ and the latus rectum is $2$ . Find the equation.

Pitfalls / Tricks

Determine which is the major axis by comparing the denominators of $x^2$ and $y^2$ , the larger denominator's axis is the major axis.
$c^2 = |a^2 - b^2|$ , always.
The focal-sum $2a$ refers to twice the semi-major axis , the largest denominator's square root times $2$ .
Insight. The closer $e$ is to $1$ , the more elongated the ellipse. At $e = 1$ it degenerates to a parabola (the focus moves to infinity along one direction).