To Prove: The equation of the given hyperbola is
It has been shown that the equation of the given hyperbola is
Given:
A hyperbola with center
Concept used:
The difference of the distances from the foci to each point on the hyperbola is constant.
Calculation:
It is given that the hyperbola has center
Then, the vertices of the hyperbola must be
Consider the vertex
So, the difference of the distances from the foci to this vertex is
It is given that the difference of the distances from the foci to each point on the hyperbola is constant.
Since the vertex is also a point on the hyperbola, it follows that the difference of the distances from the foci to each point on the hyperbola is
Now, depending on if the point is on the upper branch of the hyperbola or the lower branch of the hyperbola, the distance between that point and the upper focus is lesser than the distance between that point and the lower focus and vice versa.
Hence, it can be said that if
Now, applying distance formula,
Put these values in
Simplifying,
Squaring both sides,
Simplifying,
On further simplification,
Continuing simplification,
Squaring both sides,
Simplifying,
Put
Solving,
This shows that the equation of the given hyperbola is
Conclusion:
It has been shown that the equation of the given hyperbola is
Chapter 8 Solutions
PRECALCULUS:GRAPHICAL,...-NASTA ED.
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
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