水平椭圆平移方程式

Equation of an Ellipse

General Formula

In an ellipse, the general formula for an ellipse centered at the origin

Derivation

An ellipse is very similar two a circle. In fact, every ellipse is just a distorted circle. While a circle only has one radii, denoted $r$, an ellipse has two radii, denoted $a$ and $b$. The area of a circle is $r^2 \pi$, and the area of and ellipse is $ab \pi$. But back to the formula. Since an ellipse is so close to a circle, the derivation is too. We first start with the general formula, $x^2 + y^2 = r^2$. We divide both sides by $r^2$, giving us $\dfrac{x^2}{r^2} + \dfrac{y^2}{r^2} = 1$ Since we have two radii, $a$ and $b$, we have our final equation $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$

As you can see, we have our equation. Just like in a circle, to move the ellipse, we substitute $x$ with $(x - h)$ and $y$ with $(y - k)$. Another thing to note is that when we change the $1$ on the right side of the equation, the ellipse dilates. If we change to $4$, each side increased by a factor of $2$, and if we increase it by $9$, each side increases by a factor of $3$.

(Video) Center of an Ellipse

by Anusha Rahman

In this video by Anusha Rahman, you will learn about the center of an ellipse.

Summary

In essence, an ellipse is a stretched or compressed circle. We've learned about an ellipse in standard form,

$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

, where $(h,k)$ is the center of the ellipse, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis. The parts of the equation are fairly comparable to other function equations we've seen in the past. You can use this equation to graph an ellipse.

The equation in standard form represents an ellipse centered at the point, $(h,k)$. There are two forms of the equation: standard form and general form.

To find the equation, we can plug in these values into the standard form. To derive the standard form equation, completing the square is often used.

As a review, vertices are the endpoints of the major axis, co-vertices are the endpoints of the minor axis, and foci are two fixed points inside the ellipse that help to define its shape. The major axis is parallel to one of the coordinates axes (x- or y-axis).

Centers of Ellipses

So far, we've learned about the standards of ellipses, where the center is at the origin. However, the ellipse is dynamic, and the center can be translated to $(h,k)$.

The cartesian coordinate plane is in black, which is to the left of the major and minor axis of the ellipse. This image shows an ellipse with a center at $(h,k)$.

The equation of this ellipse is: $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

We can apply transformations to this parent equation.