SAT II Math II : Circles, Ellipses, and Hyperbolas

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

Example Question #1 : Circles, Ellipses, And Hyperbolas

Circle

Refer to the above figure. The circle has its center at the origin. What is the equation of the circle?

Possible Answers:

Correct answer:

Explanation:

The equation of a circle with center  and radius  is

The center is at the origin, or , so . To find , use the distance formula as follows:

Note that we do not actually need to find 

We can now write the equation of the circle:

Example Question #2 : Circles, Ellipses, And Hyperbolas

Circle

Refer to the above diagram. The circle has its center at the origin;  is the point . What is the length of the arc , to the nearest tenth?

Possible Answers:

Correct answer:

Explanation:

First, it is necessary to determine the radius of the circle. This is the distance between  and , so we apply the distance formula:

The circumference of the circle is 

Now we need to find the degree measure of the arc. We can do this best by examining this diagram:

Circle

The degree measure of  is also the measure  of the central angle formed by the green radii. This is found using the relationship

Using a calculator, we find that . We can adjust for the location of :

which is the degree measure of the arc.

Now we can calculate the length of the arc:

Example Question #3 : Circles, Ellipses, And Hyperbolas

On the coordinate plane, the vertices of a square are at the points with coordinates . Give the equation of the circle.

Possible Answers:

Correct answer:

Explanation:

The figure in question is below.

Incircle 1

The center of the circle can be seen to be the origin, so, if the radius is , the equation will be

.

The circle passes through the midpoints of the sides, so we will find one of these midpoints. The midpoint  of the segment with endpoints  and  can be found by using the midpoint equations, setting :

The circle passed though this midpoint . The segment from this point to the origin  is a radius, and its length is equal to . Using the following form of the distance formula, since we only need the square of the radius:

,

set :

Substituting in the circle equation for , we get the correct response,

Example Question #4 : Circles, Ellipses, And Hyperbolas

Find the diameter of the circle with the equation .

Possible Answers:

Correct answer:

Explanation:

Start by putting the equation in the standard form of the equation of a circle by completing the square. Recall the standard form of the equation of a circle:

, where the center of the circle is at  and the radius is .

From the equation, we know that .

Since the radius is , double its length to find the length of the diameter. The length of the diameter is .

Example Question #5 : Circles, Ellipses, And Hyperbolas

A triangle has its vertices at the points with coordinates , and . Give the equation of the circle that circumscribes it.

Possible Answers:

None of these

Correct answer:

Explanation:

The circumscribed circle of a triangle is the circle which passes through all three vertices of the triangle.

In general form, the equation of a circle is

.

Since the circle passes through the origin, substitute ; the equation becomes

Therefore, we know the equation of any circle passing through the origin takes the form 

for some .

Since the circle passes through , substitute ; the equation becomes

Solving for :

Now we know that the equation takes the form

 

for some .

Since the circle passes through , substitute ; the equation becomes

Solving for :

The general form of the equation of the circle is therefore

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