# Precalculus : Circles

## Example Questions

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### Example Question #27 : Determine The Equation Of A Circle In Standard Form

Write the equation for a circle with a center at (5,6) and a radius of 2.

Explanation:

The general equation for a circle is:

The coordinates of the center of the circle is (h,k) and r is the radius.

So plugging in the coordinates the circle equations is:

### Example Question #28 : Determine The Equation Of A Circle In Standard Form

Find the equation of the circle in standard form.

Explanation:

The equation of a circle in standard form is given as

where  is the coordinates of the center of the circle and  is the length of the radius.

In order to get the equation

in standard form we must complete the square.

First we group together similar variables

Then we complete the square by

1. finding half of the coefficients of each variable with degree 1
2. squaring those results
3. and then adding those numbers to the equation.

As such,

And because the square of a difference is given as this equation through factoring

we have

or

### Example Question #29 : Determine The Equation Of A Circle In Standard Form

Given the following equation for a circle, determine the coordinates of its center, as well as the coordinates of the four points directly, above, below, to the left, and to the right of the center:

Explanation:

First we must express the equation in standard form we can determine what the radius of our circle will be. The standard form for the equation of a circle is given as follows:

Where the point (h,k) gives the center of the circle and r is the radius of the circle, which can be easily determined by taking the square root of once the equation is in standard form. Our first step is to multiply both sides of the equation by 3 to cancel the division by 3 on the left side:

Now we can see that our equation is in standard form, where h=-5 and k=4, which tells us the coordinates of the center of the circle:

We can also determine the radius of the circle by taking the square root of :

Now that we know the center of the circle is at (-5,4), and that its radius is 3, we can find the points directly above and below the center by adding 3 to its y-coordinate, and then subtracting 3, respectively, giving us:

and

Similarly, to find the points directly to the left and to the right of the center, we subtract 3 from its x-coordinate, and then add 3, respectively, giving us:

and

### Example Question #30 : Determine The Equation Of A Circle In Standard Form

Graph the circle indicated by the equation

Explanation:

We must begin by recalling the general formula for the equation of a circle.

Where circle has center of coordinates  and radius of .

That means that looking at our equation, we can see that the center is .

If , then taking the square roots gives us a radius of 2.

We then look at our possible choices.  Only two are centered at .  Of these two, one has a radius of 2 while one has a radius of 4.  We want the former.

### Example Question #31 : Determine The Equation Of A Circle In Standard Form

If each mark on the graph represents  units, what is the equation of the circle?

Explanation:

Since the circle is centered at  we use the  most basic form for the equation for a circle:

.

Given the circle has a radius of  marks, which represent  units each, the circle has a radius of  units.

We then plug in  for

and simplify:

### Example Question #91 : Conic Sections

Which point is NOT on the circle defined by ?

Explanation:

The point is the center of the circle - it is not on the circle.

We can test to see if the other points are actually on the circle by plugging in their x and y values into the equation. For example, to verify that

is actually on the circle, we can plug in  for x and for y:

this is true, so that point is on the circle.

### Example Question #33 : Determine The Equation Of A Circle In Standard Form

Which best describes the point and the circle ?

The point is inside the circle

The point is on the circle

The point is outside the circle

The point is the focus of the circle

The point has no relationship with the circle