# High School Math : Conic Sections

## Example Questions

### Example Question #1 : Pre Calculus

What is the center and radius of the circle indicated by the equation?

Explanation:

A circle is defined by an equation in the format .

The center is indicated by the point  and the radius .

In the equation , the center is  and the radius is .

### Example Question #1 : Conic Sections

What is the shape of the graph indicated by the equation?

Hyperbola

Parabola

Ellipse

Circle

Ellipse

Explanation:

An ellipse has an equation that can be written in the format. The center is indicated by , or in this case .

### Example Question #3 : Pre Calculus

A conic section is represented by the following equation:

What type of conic section does this equation represent?

Parabola

Ellipse

Hyperbola

Circle

Hyperbola

Explanation:

The simplest way to know what kind of conic section an equation represents is by checking the coefficients in front of each variable. The equation must be in general form while you do this check. Luckily, this equation is already in general form, so it's easy to see. The general equation for a conic section is the following:

Assuming the term  is 0 (which it usually is):

• If A equals C, the equation is a circle.
• If A and C have the same sign (but are not equal to each other), the equation is an ellipse.
• If either A or C equals 0, the equation is a parabola.
• If A and C are different signs (i.e. one is negative and one is positive), the equation is a hyperbola.

### Example Question #4 : Pre Calculus

A conic section is represented by the following equation:

Which of the following best describes this equation?

vertical ellipse with center and a major axis length of

vertical parabola with vertex and a vertical stretch factor of

horizontal hyperbola with center and asymptotes with slopes of  and

horizontal hyperbola with center of and asymptotes with slopes of and

vertical hyperbola with center and asymptotes with slopes of  and

horizontal hyperbola with center of and asymptotes with slopes of and

Explanation:

First, we need to make sure the conic section equation is in a form we recognize. Luckily, this equation is already in standard form:

The first step is to determine the type of conic section this equation represents. Because there are two squared variables ( and ), this equation cannot be a parabola.  Because the coefficients in front of the squared variables are different signs (i.e. one is negative and the other is positive), this equation must be a hyperbola, not an ellipse.

In a hyperbola, the squared term with a positive coefficient represents the direction in which the hyperbola opens. In other words, if the term is positive, the hyperbola opens horizontally. If the term is positive, the hyperbola opens vertically. Therefore, this is a horizontal hyperbola.

The center is always found at , which in this case is .

That leaves only the asymptotes. For a hyperbola, the slopes of the asymptotes can be found by dividing  by  (remember to always put the vertical value, , above the horizontal value, ). Remember that these slopes always come in pairs, with one being positive and the other being negative.

In this case,  is 3 and  is 2, so we get slopes of and .

### Example Question #5 : Pre Calculus

Find the vertex  for a parabola with equation

Explanation:

For any parabola of the form   ,  the -coordinate of its vertex is

So here, we have

We plug this back into the original equation to find :

### Example Question #6 : Pre Calculus

What is the minimal value of

over all real numbers?

No minimum value.

Explanation:

Since this is an upwards-opening parabola, its minimum value will occur at the vertex.  The -coordinate for the vertex of any parabola of the form

is at

So here,

We plug this value back into the equation of the parabola, to find the value of the function at this

Thus the minimal value of the expression is