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 #11 : How To Find The Equation Of A Circle

A square on the coordinate plane has as its vertices the points . Give the equation of a circle inscribed in 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 equal to the sidelength of the square, which is 8, so, consequently, its radius is half this, or 4. Therefore, in the standard form of the equation, 

,

substitute  and .

Example Question #12 : How To Find The Equation Of A Circle

Give the area 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:

, and the area of the circle is 

 

Example Question #13 : How To Find The Equation Of A Circle

Which of the following is the equation of a circle with center at the origin and area  ?

Possible Answers:

Correct answer:

Explanation:

The standard form of the equation of a circle is 

,

where the center is  and the radius is .

The center of the circle is the origin, so , and the equation is 

for some .

The area of the circle is , so

We need go no further; we can substitute to get the equation .

Example Question #14 : How To Find The Equation Of A Circle

Which of the following is the equation of a circle with center at the origin and circumference  ?

Possible Answers:

None of the other responses gives the correct answer.

Correct answer:

None of the other responses gives the correct answer.

Explanation:

The standard form of the equation of a circle is 

,

where the center is  and the radius is 

 

The center of the circle is the origin, so .

The equation will be

for some .

The circumference of the circle is , so

The equation is , which is not among the responses.

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