# Precalculus : Pre-Calculus

## Example Questions

### Example Question #115 : Understand Features Of Hyperbolas And Ellipses

Find the foci of the hyperbola with the following equation:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is vertical and its center is located at .

Next, find .

The foci are then located at  and .

### Example Question #116 : Understand Features Of Hyperbolas And Ellipses

Find the foci of the hyperbola with the following equation:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is vertical and its center is located at .

Next, find .

The foci are then located at  and .

### Example Question #211 : Conic Sections

Find the foci of the hyperbola with the following equation:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is vertical and its center is located at .

Next, find .

The foci are then located at  and .

### Example Question #212 : Conic Sections

Find the foci for the hyperbola with the following equation:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is vertical and its center is located at .

Next, find .

The foci are then located at  and .

### Example Question #213 : Conic Sections

Which point is one of the foci of the hyperbola ?

Explanation:

To find the foci of a hyperbola, first determine a and b, and then use the relationship

In this case, the major axis is horizontal since x comes first, so and .

Solve for c: add 9 to both sides

take the square root

Since the center is and the major axis is the horizontal one, our foci are . The only choice that works is

.

### Example Question #214 : Conic Sections

Determine the length of the foci for the following hyperbola equation:

Explanation:

To solve, simply use the follow equation where c is the length of the foci.

In this particular case,

Thus,

### Example Question #215 : Conic Sections

Find the foci of the hyperbola with the following equation:

and

and

and

and

and

and

Explanation:

The standard form of the equation for a hyperbola is given by

The foci are located at (h+c, k) and (h-c, k), where c is found by using the formula

Since our equation is already in standard form, you can see that

Plugging into the formula

So the foci are found at

AND

### Example Question #216 : Conic Sections

How can this graph be changed to be the graph of

?

The graph should be an ellipse and not a hyperbola.

The center box should extend up to  and down to , stretching the graph.

The -intercepts should be at the points and .

The graph should have -intercepts and not -intercepts.

The -intercepts should be at the points and .

The -intercepts should be at the points and .

Explanation:

This equation should be thought of as .

This means that the hyperbola will be determined by a box with x-intercepts at and y-intercepts at .

The hyperbola was incorrectly drawn with the intercepts at instead.

### Example Question #121 : Hyperbolas And Ellipses

Which of the following would NOT be true of the graph for ?

All of these statements are true.

The graph never intersects with the -axis.

The graph is centered at .

The graph opens up and down.

The graph never intersects with the -axis.

The graph never intersects with the -axis.

Explanation:

The graph should look like this:

### Example Question #211 : Conic Sections

Which of these equations produce this graph, rotated 90 degrees?

Explanation:

Rotated 90 degrees, this graph would be opening up and down instead of left and right, so the equation will have the y term minus the x term.

The box that the hyperbola is drawn around will also rotate. It will now be up/down 2 and left/right 3.

This makes the correct equation

.