### All SAT II Math II Resources

## Example Questions

### Example Question #1 : Coordinate Geometry

Which of the following equations represent a parabola?

**Possible Answers:**

**Correct answer:**

The parabola is represented in the form . If there is a variable in the denominator or as an exponent, it is not a parabola.

The only equation that has an order of two is:

### Example Question #1 : Circles, Ellipses, And Hyperbolas

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

**Possible Answers:**

**Correct answer:**

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 #1 : Circles, Ellipses, And Hyperbolas

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:**

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:

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 #1 : 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:**

The figure in question is below.

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 #1 : Coordinate Geometry

Find the diameter of the circle with the equation .

**Possible Answers:**

**Correct answer:**

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 #1 : 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:**

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

### Example Question #1 : Coordinate Geometry

Which of the following symmetries applies to the graph of the relation

?

I) Symmetry with respect to the origin

II) Symmetry with respect to the -axis

III) Symmetry with respect to the -axis

**Possible Answers:**

II only

III only

None of these

I only

I, II, and III

**Correct answer:**

II only

The relation

is a circle with center and radius .

In other words, it is a circle with center at the origin, translated right units and up units.

or

is a circle translated right 4 units and up zero units. The upshot is that the circle moves along the -axis only, and therefore is symmetric with respect to the -axis, but not the -axis. Also, as a consequence, it is not symmetric with respect to the origin.

### Example Question #132 : Geometry

Refer to the above figure.

Which of the following functions is graphed?

**Possible Answers:**

The correct answer is not given among the other responses.

**Correct answer:**

Below is the graph of :

The given graph is the graph of shifted 6 units left (that is, unit right) and 3 units up.

The function graphed is therefore

where . That is,

### Example Question #133 : Geometry

Refer to the above figure.

Which of the following functions is graphed?

**Possible Answers:**

**Correct answer:**

Below is the graph of :

If the graph of is translated by shifting each point to the point , the graph of

is formed. If the graph is then shifted upward by three units, the new graph is

Since the starting graph was , the final graph is

, or,

### Example Question #131 : Geometry

Refer to the above figure.

Which of the following functions is graphed?

**Possible Answers:**

**Correct answer:**

Below is the graph of :

If the graph of is translated by shifting each point to the point , the graph of

is formed. If the graph is then shifted right by four units, the new graph is

Since the starting graph was , the final graph is

, or

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