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:

Recall that the solution of a system of equations is given by the intersection points of the graphs. Thus, this question is really asking how many times a parabola and a line can intersect. Visualize a parabola and a line. For this purpose, let's say that the parabola is facing up. If a line is drawn horizontally under the vertex, then it would not intersect the parabola at all, so the system would have no solutions. If the line is tangent to, or just skims the edge, of the parabola then it would only intersect once and the system would have one solution. If the line goes straight through the parabola, then it would intersect twice. There is no other option for the orientation of the line and parabola. Thus, there cannot be three solutions. 

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 .

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 

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 .

To solve, set the two quadratics equal to each other and then combine like terms:

subtract everything on the right from both sides to combine like terms.

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

To find the y-coordinates, plug these into either equation:

To solve, first re-write the second one so that y is isolated on the left side:

Now set the two quadratics equal to each other:

add/subtract all of the terms from the right side so that this is a quadratic equal to zero.

combine like terms.

Using the quadratic formula or by factoring, we get the two solutions and .

To get the y-coordinates, plug these numbers into either function:

To solve, set the two polynomials equal to each other:

add/subtract all of the terms from the right side from both sides.

combine like terms.

Solving with the quadratic formula or by factoring gives us the solutions 5 and -3.

To get the y-coordinates, plug these numbers into either of the original equations:

Recall that the polar equations of conic sections can come in the following forms:

, where  is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

 will give an ellipse.

 will give a parabola.

 will give a hyperbola.

Now, for the given conic section,  so it must be a hyperbola.

Recall that the polar equations of conic sections can come in the following forms:

, where  is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

 will give an ellipse.

 will give a parabola.

 will give a hyperbola.

Now, for the given conic section,  so it must be an ellipse.