Precalculus : Circles

Example Questions

Example Question #7 : Determine The Equation Of A Circle In Standard Form

Which choice would be the circle in standard form?

Explanation:

To find the standard form of this equation, we have to complete the square for both x and y. It's easiest to do this if we group together the y terms first, then the left terms, and subtract the constant from both sides:

original: subtract 1 from both sides; group x and y

To make the y terms into a square, we have to add 1, since half of 2 is 1, and .

To make the x terms into a square, we have to add 9, since half of -6 is -3, and :

Now we just have to re-write the y and x terms as the squares that they are, and simplify the right side:

Example Question #8 : Determine The Equation Of A Circle In Standard Form

Give the center and radius for the circle .

Explanation:

To determine the center and radius, put this equation into standard form. Standard form is , where is the center and is the radius.

First, group the y terms together and the x terms together, and subtract 37 from both sides:

We're leaving a space after these terms because we're completing the square. To make the y terms into a square, we need to add 49, since half of -14 is -7, and

To make the x terms into a square, we need to add 4, since half of -4 is -2, and

We'll need to add these to both sides to keep both sides equal:

Now we can re-write the quadratics on the left as squares, and simplify the right side:

Interpreting this equation, we can see that the center would be at and the radius is

Example Question #9 : Determine The Equation Of A Circle In Standard Form

Write the equation for in standard form.

Explanation:

To write this in standard form, we will have to complete the square for both x and y. To do this more easily, group the x terms, then the y terms, and then subtract 43 from both sides:

To complete the square for x, we have to add to both sides; to complete the square for y, we have to add to both sides:

Condense the left side and add together the numbers on the right:

Example Question #1 : Determine The Equation Of A Circle In Standard Form

Write the standard equation of the circle.

Explanation:

Group the x and y terms on one side of the equation and the constant on the other.

Complete the square by taking half of the middle number for each variable and squaring it. Add the number to the other side of the equation.

Example Question #11 : Determine The Equation Of A Circle In Standard Form

Find the equation of the circle if it is centered at  and has a radius of  units.

Explanation:

The equation of a circle centered at   with radius  units in standard form is

For the circle ceentered at  with radius  units has the equation

Example Question #12 : Determine The Equation Of A Circle In Standard Form

Find the equation of the circle if it is centered at  and has a radius of  units.

Explanation:

The equation of a circle centered at   with radius  units in standard form is

For the circle ceentered at  with radius  units has the equation

or

Example Question #13 : Determine The Equation Of A Circle In Standard Form

is a point on a circle whose center is at . What is the standard equation of this circle?

Explanation:

The standard form of the equation of a circle is

where the center of the circle resides at the point .

Given the center of a circle and a point on the rim of the circle, one can use the distance formula to find the radius.

Now plug in the point for the center and radius into the standard equation of a circle:

Example Question #14 : Determine The Equation Of A Circle In Standard Form

Rewrite the following in standard form:

Explanation:

To solve for general form, we want to format the given equation as

Example Question #15 : Determine The Equation Of A Circle In Standard Form

Rewrite the following equation in standard form:

Explanation:

To rewrite in standard form, we must format the equation as

Example Question #16 : Determine The Equation Of A Circle In Standard Form

Rewrite the given equation in standard form: