Ellipse

What is an Ellipse?

Ever seen the shape of planets orbiting the sun? Or the shadow of a circle when viewed from the side? Well, shapes like those are called ellipses! An ellipse isn't just a "flattened" circle, but there's a cool mathematical definition behind it.

So here's the thing, an ellipse is a collection of points where the sum of distances to two specific points is always the same. These two specific points are called foci. Just imagine you have two nails and a string. If you tie the string to both nails, then pull a pencil until the string is tight and draw a complete curve, the curve formed is an ellipse!

From the visualization above, notice point $P$. The distance from $P$ to focus $F_1$ (which we call $r_1$) plus the distance from $P$ to focus $F_2$ (which we call $r_2$) will always be the same for all points on the ellipse. This is the fundamental characteristic of an ellipse!

Ellipse Components

Before we discuss the formulas, let's get familiar with the important parts of an ellipse. Each part has its own role in determining the shape of the ellipse.

Here are the components you need to know:

1. Ellipse center is the midpoint of the ellipse, usually written with letter $O$. All measurements in the ellipse refer to this point.

2. Foci ($F_1$ and $F_2$) are two fixed points that serve as reference for the ellipse definition. The distance between the two foci is called the focal distance.

3. Major axis is the longest line that passes through the ellipse center and both foci. Its endpoints are points $A_1$ and $A_2$.

4. Minor axis is the shortest line that passes through the ellipse center and is perpendicular to the major axis. Its endpoints are points $B_1$ and $B_2$.

5. Semi-major ($a$) is half the length of the major axis, which is the distance from center to the major axis endpoint.

6. Semi-minor ($b$) is half the length of the minor axis, which is the distance from center to the minor axis endpoint.

Remember, in an ellipse we always have $a > b$. If $a = b$, the shape becomes a circle!

Ellipse Equations

Now, let's get into the fun part: how to write an ellipse in mathematical equation form. There are several forms depending on position and orientation.

Center at Origin

If the ellipse center is at $O(0,0)$, there are two possible orientations:

When the major axis is parallel to the $X$ axis (horizontal), the ellipse equation is:

with the condition $a > b$.

When the major axis is parallel to the $Y$ axis (vertical), the ellipse equation is:

with the condition $a > b$.

Shifted Center

If the ellipse center is not at the origin, but at point $(h, k)$, the equation becomes:

Visualization:

Important Relationships

There's a formula that always applies to every ellipse:

where $c$ is the distance from center to focus.

Eccentricity of an ellipse is defined as:

The eccentricity value of an ellipse is always $0 < e < 1$. The closer to $0$, the more circular the ellipse becomes. The closer to $1$, the more elongated the ellipse becomes.

Exercises

1. Find the equation of an ellipse with center at $(0,0)$, major axis length $12$ and minor axis length $8$, with horizontal major axis.

2. Given ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$. Find the coordinates of the foci and the eccentricity of the ellipse.

3. An ellipse has center at $(3, -1)$, foci at $(3, 2)$ and $(3, -4)$, and minor axis length $6$. Find the equation of the ellipse.

4. Find the equation of an ellipse that passes through points $(4, 3)$ and $(6, 2)$ with center at $(0, 0)$ and horizontal major axis.

Answer Key

1. Solution:

   Given:

   • Center at $(0,0)$
   • Major axis length = $12$, so $2a = 12$, thus $a = 6$
   • Minor axis length = $8$, so $2b = 8$, thus $b = 4$
   • Horizontal major axis

   Ellipse equation with horizontal major axis:

   $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

   Substituting values $a = 6$ and $b = 4$:

   $$\frac{x^2}{36} + \frac{y^2}{16} = 1$$

2. Solution:

   From equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$:

   • $a^2 = 25$, so $a = 5$
   • $b^2 = 16$, so $b = 4$

   Since $a^2 > b^2$, the major axis is horizontal.

   Calculate $c$:

   $$c^2 = a^2 - b^2 = 25 - 16 = 9$$

   $$c = 3$$

   Foci coordinates: $(\pm 3, 0)$ which are $(-3, 0)$ and $(3, 0)$

   Eccentricity:

   $$e = \frac{c}{a} = \frac{3}{5} = 0{.}6$$

3. Solution:

   Given:

   • Center: $(3, -1)$
   • Foci: $(3, 2)$ and $(3, -4)$
   • Minor axis length = $6$, so $2b = 6$, thus $b = 3$

   Since the foci have the same $x$ coordinate ($x = 3$), the major axis is vertical.

   $$\text{Distance between foci} = |2 - (-4)| = 6$$

   so $2c = 6$, thus $c = 3$

   Calculate $a$:

   $$c^2 = a^2 - b^2$$

   $$9 = a^2 - 9$$

   $$a^2 = 18$$

   $$a = 3\sqrt{2}$$

   Ellipse equation with center $(h,k) = (3,-1)$ and vertical major axis:

   $$\frac{(x-3)^2}{9} + \frac{(y+1)^2}{18} = 1$$

4. Solution:

   An ellipse with center $(0,0)$ and horizontal major axis has the equation:

   $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

   Substituting point $(4,3)$:

   $$\frac{16}{a^2} + \frac{9}{b^2} = 1 \quad \text{...(1)}$$

   Substituting point $(6,2)$:

   $$\frac{36}{a^2} + \frac{4}{b^2} = 1 \quad \text{...(2)}$$

   Let $u = \frac{1}{a^2}$ and $v = \frac{1}{b^2}$, then:

   $$16u + 9v = 1 \quad \text{...(1)}$$

   $$36u + 4v = 1 \quad \text{...(2)}$$

   From equation (1): $v = \frac{1-16u}{9}$

   Substituting into equation (2):

   $$36u + 4 \cdot \frac{1-16u}{9} = 1$$

   $$36u + \frac{4-64u}{9} = 1$$

   $$324u + 4 - 64u = 9$$

   $$260u = 5$$

   $$u = \frac{1}{52}$$

   So $a^2 = 52$

   Substituting back:

   $$v = \frac{1-16 \cdot \frac{1}{52}}{9} = \frac{1-\frac{16}{52}}{9} = \frac{\frac{36}{52}}{9} = \frac{4}{52} = \frac{1}{13}$$

   So $b^2 = 13$

   Ellipse equation: $\frac{x^2}{52} + \frac{y^2}{13} = 1$