# Algebra Examples

Find the Ellipse: Center (0,1), Focus (6,1), Vertex (8,1)
, ,
Step 1
There are two general equations for an ellipse.
Horizontal ellipse equation
Vertical ellipse equation
Step 2
is the distance between the vertex and the center point .
Step 2.1
Use the distance formula to determine the distance between the two points.
Step 2.2
Substitute the actual values of the points into the distance formula.
Step 2.3
Simplify.
Step 2.3.1
Subtract from .
Step 2.3.2
Raise to the power of .
Step 2.3.3
Subtract from .
Step 2.3.4
Raising to any positive power yields .
Step 2.3.5
Step 2.3.6
Rewrite as .
Step 2.3.7
Pull terms out from under the radical, assuming positive real numbers.
Step 3
is the distance between the focus and the center .
Step 3.1
Use the distance formula to determine the distance between the two points.
Step 3.2
Substitute the actual values of the points into the distance formula.
Step 3.3
Simplify.
Step 3.3.1
Subtract from .
Step 3.3.2
Raise to the power of .
Step 3.3.3
Subtract from .
Step 3.3.4
Raising to any positive power yields .
Step 3.3.5
Step 3.3.6
Rewrite as .
Step 3.3.7
Pull terms out from under the radical, assuming positive real numbers.
Step 4
Using the equation . Substitute for and for .
Step 4.1
Rewrite the equation as .
Step 4.2
Raise to the power of .
Step 4.3
Raise to the power of .
Step 4.4
Move all terms not containing to the right side of the equation.
Step 4.4.1
Subtract from both sides of the equation.
Step 4.4.2
Subtract from .
Step 4.5
Divide each term in by and simplify.
Step 4.5.1
Divide each term in by .
Step 4.5.2
Simplify the left side.
Step 4.5.2.1
Dividing two negative values results in a positive value.
Step 4.5.2.2
Divide by .
Step 4.5.3
Simplify the right side.
Step 4.5.3.1
Divide by .
Step 4.6
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 4.7
Simplify .
Step 4.7.1
Rewrite as .
Step 4.7.1.1
Factor out of .
Step 4.7.1.2
Rewrite as .
Step 4.7.2
Pull terms out from under the radical.
Step 4.8
The complete solution is the result of both the positive and negative portions of the solution.
Step 4.8.1
First, use the positive value of the to find the first solution.
Step 4.8.2
Next, use the negative value of the to find the second solution.
Step 4.8.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 5
is a distance, which means it should be a positive number.
Step 6
The slope of the line between the focus and the center determines whether the ellipse is vertical or horizontal. If the slope is , the graph is horizontal. If the slope is undefined, the graph is vertical.
Step 6.1
Slope is equal to the change in over the change in , or rise over run.
Step 6.2
The change in is equal to the difference in x-coordinates (also called run), and the change in is equal to the difference in y-coordinates (also called rise).
Step 6.3
Substitute in the values of and into the equation to find the slope.
Step 6.4
Simplify.
Step 6.4.1
Simplify the numerator.
Step 6.4.1.1
Multiply by .
Step 6.4.1.2
Subtract from .
Step 6.4.2
Simplify the denominator.
Step 6.4.2.1
Multiply by .
Step 6.4.2.2
Subtract from .
Step 6.4.3
Divide by .
Step 6.5
The general equation for a horizontal ellipse is .
Step 7
Substitute the values , , , and into to get the ellipse equation .
Step 8
Simplify to find the final equation of the ellipse.
Step 8.1
Simplify the numerator.
Step 8.1.1
Multiply by .
Step 8.1.2
Step 8.2
Raise to the power of .
Step 8.3
Multiply by .
Step 8.4
Simplify the denominator.
Step 8.4.1
Apply the product rule to .
Step 8.4.2
Raise to the power of .
Step 8.4.3
Rewrite as .
Step 8.4.3.1
Use to rewrite as .
Step 8.4.3.2
Apply the power rule and multiply exponents, .
Step 8.4.3.3
Combine and .
Step 8.4.3.4
Cancel the common factor of .
Step 8.4.3.4.1
Cancel the common factor.
Step 8.4.3.4.2
Rewrite the expression.
Step 8.4.3.5
Evaluate the exponent.
Step 8.5
Multiply by .
Step 9