Algebra Examples

Find Hyperbola: Center (2,3), Focus (-4,3), Vertex (1,3)
, ,
There are two general equations for a hyperbola.
Horizontal hyperbola equation
Vertical hyperbola equation
is the distance between the vertex and the center point .
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Use the distance formula to determine the distance between the two points.
Substitute the actual values of the points into the distance formula.
Simplify.
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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 .
Add and to get .
Any root of is .
is the distance between the focus and the center .
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Use the distance formula to determine the distance between the two points.
Substitute the actual values of the points into the distance formula.
Simplify.
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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 .
Add and to get .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Using the equation . Substitute for and for to get .
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Move to the right side of the equation by subtracting from both sides of the equation.
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
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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 hyperbola is vertical or horizontal. If the slope is , the graph is horizontal. If the slope is undefined, the graph is vertical.
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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.
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Simplify the numerator.
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Multiply by to get .
Subtract from to get .
Simplify the denominator.
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Remove parentheses.
Multiply by to get .
Add and to get .
Divide by to get .
The general equation for a horizontal hyperbola is .
Substitute the values , , , and into to get the hyperbola equation .
Simplify to find the final equation of the hyperbola.
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Multiply by to get .
Simplify the denominator.
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Remove parentheses around .
One to any power is one.
Divide by to get .
Multiply by to get .
Simplify the denominator.
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Remove parentheses around .
Rewrite as .
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