The ellipse: two definitions and the canonical equation

An ellipse is the locus of all the points in the plane whose distances from two given points, the foci, add up to a constant. The constant is called $\displaystyle 2a$, and it is greater than the distance between the foci.

We choose the foci on the $\displaystyle x$-axis, symmetric about the $\displaystyle y$-axis: $\displaystyle F_1(-c,0)$ and $\displaystyle F_2(c,0)$. The condition:

The working below isolates one root and squares twice:

• $\displaystyle {\sqrt{(x+c)^{2}+y^{2}}}={2a-\sqrt{(x-c)^{2}+y^{2}}}$, squared: $\displaystyle {(x+c)^{2}}={4a^{2}-4a\sqrt{(x-c)^{2}+y^{2}}+(x-c)^{2}}$.
• $\displaystyle {4cx-4a^{2}}={-4a\sqrt{(x-c)^{2}+y^{2}}}$, that is $\displaystyle {a\sqrt{(x-c)^{2}+y^{2}}}={a^{2}-cx}$.
• Squared again: $\displaystyle {a^{2}x^{2}-2a^{2}cx+a^{2}c^{2}+a^{2}y^{2}}={a^{4}-2a^{2}cx+c^{2}x^{2}}$, so $\displaystyle {(a^{2}-c^{2})x^{2}+a^{2}y^{2}}={a^{2}(a^{2}-c^{2})}$.
• Write $\displaystyle {b^{2}}={a^{2}-c^{2}}$ (positive, since $\displaystyle a>c$) and divide by $\displaystyle a^{2}b^{2}$.

This gives the canonical equation of the ellipse:

From the equation we see:

• Symmetry – about both axes and the origin: $\displaystyle x$ and $\displaystyle y$ appear only squared.
• The vertices – where it meets the axes: $\displaystyle (\pm a,0)$ and $\displaystyle (0,\pm b)$.
• The major axis – the segment from $\displaystyle (-a,0)$ to $\displaystyle (a,0)$, of length $\displaystyle 2a$; the minor axis – from $\displaystyle (0,-b)$ to $\displaystyle (0,b)$, of length $\displaystyle 2b$. $\displaystyle a$ is the semi-major axis and $\displaystyle b$ the semi-minor axis.
• The foci – on the major axis, $\displaystyle c$ from the origin.
• The vertex $\displaystyle (0,b)$ is $\displaystyle a$ away from each focus: $\displaystyle {b^{2}+c^{2}}={a^{2}}$.

An example: $\displaystyle {\frac{x^{2}}{25}+\frac{y^{2}}{9}}={1}$: $\displaystyle {a}={5}$, $\displaystyle {b}={3}$, $\displaystyle {c}={\sqrt{25-9}}={4}$. The foci are $\displaystyle (\pm4,0)$, and every point’s distances from them add up to $\displaystyle 10$.

When $\displaystyle b>a$ the major axis is on the $\displaystyle y$-axis, and the foci are $\displaystyle (0,\pm c)$ with $\displaystyle {c}={\sqrt{b^{2}-a^{2}}}$.

Another definition: an ellipse is a circle squashed (or stretched) in one direction.

• Squash the circle $\displaystyle {x^{2}+y^{2}}={a^{2}}$ towards the $\displaystyle x$-axis: each of its points $\displaystyle (x,Y)$ goes to $\displaystyle (x,y)$ with $\displaystyle {y}={\frac{b}{a}Y}$.
• $\displaystyle {Y}={\frac{a}{b}y}$, and $\displaystyle (x,Y)$ is on the circle: $\displaystyle {x^{2}+\frac{a^{2}}{b^{2}}y^{2}}={a^{2}}$.
• Divide by $\displaystyle a^{2}$: $\displaystyle {\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}}={1}$ – the same ellipse.

In the drawing, the circle $\displaystyle {x^{2}+y^{2}}={25}$ is squashed by $\displaystyle \frac{3}{5}$ towards the $\displaystyle x$-axis.

The ellipse: $\displaystyle {\frac{x^{2}}{100}+\frac{y^{2}}{36}}={1}$.

What are the vertices, the foci and the lengths of the axes?

An ellipse’s foci are $\displaystyle (\pm3,0)$, and every point’s distances from the foci add up to $\displaystyle 10$.
Write the equation of the ellipse.

The ellipse: $\displaystyle {4x^{2}+9y^{2}}={36}$.

Write the equation in canonical form. What are $\displaystyle a$ and $\displaystyle b$?

On which axis are the foci of the ellipse $\displaystyle {\frac{x^{2}}{16}+\frac{y^{2}}{25}}={1}$, and what are their coordinates?

The circle $\displaystyle {x^{2}+y^{2}}={36}$ was squashed towards the $\displaystyle x$-axis by $\displaystyle \frac{1}{3}$: each point’s $\displaystyle y$-coordinate was multiplied by $\displaystyle \frac{1}{3}$.
Write the equation of the ellipse obtained.

The Earth goes round the Sun in an ellipse, with the Sun at one focus.
The Earth’s least distance from the Sun is $\displaystyle 147.1$ million km.
The greatest distance is $\displaystyle 152.1$ million km.