# GRE Subject Test: Math : Conic Sections

## Example Questions

### Example Question #1 : Circles

What is the equation of a circle with center at  and a radius of ?

Explanation:

Step 1: Recall the general equation for a circle (if the vertex is not at :

, where center=

Step 2: Recall the shift of the graph..

If the value of  is positive, it will be shown as a negative shift in the equation.
If the value of  is negative, it will be shown as a positive shift in the equation.
If the value of  is positive, it will be shown as a negative shift in the equation.
If the value of  is negative, it will be shown as a positive shift in the equation.

Step 3: Look at the center given in the problem and find the rule(s) in step 2 that will apply:

Center=

Step 4: Plug in  into the equation of a circle:

Simplify:

### Example Question #1 : Conic Sections

What is the vertex of the equation of a circle:

Explanation:

Step 1: There are no numbers next to  and , so their is no movement of the vertex..

Step 2: Recall the vertex of a circle that does not move...

The vertex of this circle is .

### Example Question #44 : Graphs

Using the information below, determine the equation of the hyperbola.

Foci:  and

Eccentricity:

Explanation:

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci =

Distance between vertices =

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 12, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 4, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 12, we can conclude that

Center point:

Thus, the equation of the hyperbola is:

### Example Question #1 : Hyperbolas

Using the information below, determine the equation of the hyperbola.

Foci:  and

Eccentricity:

Explanation:

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci =

Distance between vertices =

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 8, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 8, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 8, we can conclude that

Center point:

Thus, the equation of the hyperbola is:

### Example Question #2 : Solve A System Of Quadratic Equations

Find the coordinate of intersection, if possible:   and .

Explanation:

To solve for x and y, set both equations equal to each other and solve for x.

Substitute  into either parabola.

The coordinate of intersection is .

### Example Question #1 : Solve A System Of Quadratic Equations

Find the intersection(s) of the two parabolas:

Explanation:

Set both parabolas equal to each other and solve for x.

Substitute both values of  into either parabola and determine .

The coordinates of intersection are:

and

### Example Question #1 : Solve A System Of Quadratic Equations

Find the points of intersection:

;

Explanation:

To solve, set both equations equal to each other:

To solve as a quadratic, combine like terms by adding/subtracting all three terms from the right side to the left side:

This simplifies to

Solving by factoring or the quadratic formula gives the solutions and .

Plugging each into either original equation gives us:

Our coordinate pairs are and .