# Algebra Examples

Find Ellipse: Center (5,3), Focus (4,3), Vertex (-5,3)
, ,
There are two general equations for an ellipse.
Horizontal ellipse equation
Vertical ellipse equation
is the distance between the vertex and the center point .
Use the distance formula to determine the distance between the two points.
Substitute the actual values of the points into the distance formula.
Simplify.
Multiply by to get .
Subtract from to get .
Raise to the power of to get .
Multiply by to get .
Subtract from to get .
Remove parentheses around .
Raising to any positive power yields .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
is the distance between the focus and the center .
Use the distance formula to determine the distance between the two points.
Substitute the actual values of the points into the distance formula.
Simplify.
Multiply by to get .
Subtract from to get .
Raise to the power of to get .
Multiply by to get .
Subtract from to get .
Remove parentheses around .
Raising to any positive power yields .
Any root of is .
Using the equation . Substitute for and for to get .
Move to the right side of the equation by subtracting from both sides of the equation.
Multiply each term in by
Multiply each term in by .
Simplify .
Multiply by to get .
Multiply by to get .
Multiply by to get .
Take the root of both sides of the to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
is a distance, which means it should be a positive number.
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.
Slope is equal to the change in over the change in , or rise over run.
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).
Substitute in the values of and into the equation to find the slope.
Simplify.
Simplify the numerator.
Multiply by to get .
Subtract from to get .
Simplify the denominator.
Remove parentheses.
Multiply by to get .
Subtract from to get .
Divide by to get .
The general equation for a horizontal ellipse is .
Substitute the values , , , and into to get the ellipse equation .
Simplify to find the final equation of the ellipse.
Multiply by to get .
Split the fraction into multiple fractions.
Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Split the fraction into multiple fractions.
Apply the product rule to .
Raise to the power of to get .
Rewrite as .
Multiply by to get .

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