# Focus

As we get further into algebra, we will start to deal with problems involving focus. But what exactly is a focus? How is this concept related to algebra, and what can it teach us about math? Let''s find out:

## What is a focus of a circle?

The concept of a focus is related to conic sections and circles. A circle has one focus, and it exists at its center. Take a look:

As we can see, the focus of this circle exists not only at its center but also at the origin of the plane. We might say that a circle is *determined* by its focus, and that it is the set of all points in a plane at a given distance from its focus.

Fun fact: The plural of focus is "foci."

## The focus of a parabola

As we might recall, a parabola is a type of conic section. Although we see parabolas in two dimensions on a plane, it is equivalent to the cross-section of a three-dimensional cone. The focus exists at the center of the conic section just as it exists at the center of a circle. We might also say that the focus represents the "point" of a conic section since that also exists at its center.

But what exactly does the focus of a parabola look like? Take a look:

As we can see, there is also a directrix on this graph that takes the form of a straight line. The vertex is the highest *or *lowest point of the line, depending on the equation. In this case, the vertex is at the lowest point of the line -- also known as the "trough." These two terms are related to the focus because the distance between the vertex and directrix is equal to the distance between the vertex and the focus.

## The foci of an ellipse

Another type of conic section is called an "ellipse." This type of conic section has not one but *two *foci. Let''s see what this looks like:

As we can see, an ellipse is not a perfect circle, but rather a circle that has been stretched or compressed into more of an oval shape. If we look at any point on the ellipse, we see that the *sum* of the distances to each focus remains constant.

## The foci of a parabola

A hyperbola also has two foci. This conic section represents the cross-section of two inverted cones, and it has two characteristic bows or arms. Let''s see what the foci of a parabola look like:

If we look at any point on either of the two arms, we can see that the *difference* between the distances to each focus is constant.

## Topics related to the Focus

Finding the Equation of a Parabola given Focus and Directrix

## Flashcards covering the Focus

## Practice tests covering the Focus

College Algebra Diagnostic Tests

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