Circles and ellipses are both shapes you can find in math, but they are not the same. Here’s how they differ:
What They Are:
A circle is all the points that are the same distance from a fixed middle point called the center.
The regular formula for a circle looks like this:
[(x - h)^2 + (y - k)^2 = r^2]
Here, ((h, k)) is the center, and (r) is how far the edge is from the center.
An ellipse is made up of points where the total distance to two fixed points (called foci) stays the same.
The formula for an ellipse is:
[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 ]
Again, ((h, k)) is the center. The letter (a) is for the longer distance from the center (semi-major axis), and (b) is for the shorter distance (semi-minor axis).
Shapes:
Circles are perfectly round.
They have the same distance from the center no matter which direction you go.
Ellipses are stretched out.
Their shape depends on the relationship between the two axes, and there's a formula to describe it: (c^2 = a^2 - b^2), where (c) is the distance from the center to each focus.
Foci:
A circle has one focus, which is its center point.
An ellipse has two foci. The distance from the center to each focus is noted as (c).
Knowing these differences helps us understand how circles and ellipses are special and unique in math!
Circles and ellipses are both shapes you can find in math, but they are not the same. Here’s how they differ:
What They Are:
A circle is all the points that are the same distance from a fixed middle point called the center.
The regular formula for a circle looks like this:
[(x - h)^2 + (y - k)^2 = r^2]
Here, ((h, k)) is the center, and (r) is how far the edge is from the center.
An ellipse is made up of points where the total distance to two fixed points (called foci) stays the same.
The formula for an ellipse is:
[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 ]
Again, ((h, k)) is the center. The letter (a) is for the longer distance from the center (semi-major axis), and (b) is for the shorter distance (semi-minor axis).
Shapes:
Circles are perfectly round.
They have the same distance from the center no matter which direction you go.
Ellipses are stretched out.
Their shape depends on the relationship between the two axes, and there's a formula to describe it: (c^2 = a^2 - b^2), where (c) is the distance from the center to each focus.
Foci:
A circle has one focus, which is its center point.
An ellipse has two foci. The distance from the center to each focus is noted as (c).
Knowing these differences helps us understand how circles and ellipses are special and unique in math!