SSAT Upper Level Math : How to find the equation of a circle

Study concepts, example questions & explanations for SSAT Upper Level Math

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Example Questions

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Example Question #1 : Circles

Circle

Give the equation of the above circle.

Possible Answers:

None of the other choices is correct.

Correct answer:

Explanation:

A circle with center  and radius  has equation

The circle has center  and radius 5, so substitute:

Example Question #1 : Circles

A circle on the coordinate plane has a diameter whose endpoints are  and . Give its equation.

Possible Answers:

Correct answer:

Explanation:

A circle with center  and radius  has equation

The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:

Therefore,  and 

The radus is the distance between the center and one endpoint, so take advantage of the distance formula using  and . We will concern ourcelves with finding the square of the radius :

Substitute: 

Example Question #3 : Circles

Circle

Give the equation of the above circle.

Possible Answers:

Correct answer:

Explanation:

A circle with center  and radius  has equation

The circle has center  and radius 4, so substitute:

Example Question #4 : Circles

A circle on the coordinate plane has a diameter whose endpoints are  and . Give its equation.

Possible Answers:

Correct answer:

Explanation:

A circle with center  and radius  has equation

The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:

Therefore,  and .

The radus is the distance between the center and one endpoint, so take advantage of the distance formula using  and . We will concern ourcelves with finding the square of the radius :

Substitute: 

Expand:

Example Question #2 : Circles

A circle on the coordinate plane has center  and circumference . Give its equation.

Possible Answers:

Correct answer:

Explanation:

A circle with center  and radius  has equation

The center is , so .

To find , use the circumference formula:

Substitute:

Example Question #6 : Circles

A circle on the coordinate plane has center  and area . Give its equation.

Possible Answers:

Correct answer:

Explanation:

A circle with center  and radius  has the equation

The center is , so .

The area is , so to find , use the area formula:

The equation of the line is therefore: 

Example Question #3 : Circles

What is the equation of a circle that has its center at  and has a radius of ?

Possible Answers:

Correct answer:

Explanation:

The general equation of a circle with center  and radius  is:

Now, plug in the values given by the question:

Example Question #8 : Circles

If the center of a circle with a diameter of 5 is located at , what is the equation of the circle?

Possible Answers:

Correct answer:

Explanation:

Write the formula for the equation of a circle with a given point, .

The radius of the circle is half the diameter, or .

Substitute all the values into the formula and simplify.

Example Question #9 : Circles

Give the circumference of the circle on the coordinate plane whose equation is

Possible Answers:

Correct answer:

Explanation:

The standard form of the equation of a circle is 

where  is the radius of the circle.

We can rewrite the equation we are given, which is in general form, in this standard form as follows:

Complete the squares. Since  and , we do this as follows:

, so , and the circumference of the circle is 

Example Question #10 : Circles

A square on the coordinate plane has as its vertices the points . Give the equation of a circle circumscribed about the square.

Possible Answers:

Correct answer:

Explanation:

Below is the figure with the circle and square in question:

Circle on axes

The center of the inscribed circle coincides with that of the square, which is the point . Its diameter is the length of a diagonal of the square, which is  times the sidelength 6 of the square - this is . Its radius is, consequently, half this, or . Therefore, in the standard form of the equation, 

,

substitute  and .

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