Question

Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (−4, 0), (4, 0); vertices:(−3, 0), (3, 0)

Accepted Answer

Identify the center of the hyperbola by finding the midpoint of the vertices. Since the vertices are at (−3, 0) and (3, 0), the center is at the midpoint: $$\left( \frac{-3 + 3}{2}, \frac{0 + 0}{2} \right). $$Determine the orientation of the hyperbola. Because the vertices and foci lie on the x-axis, the transverse axis is horizontal, so the standard form will be $$\frac{(x - h)^2$}{a^2$} - \frac{(y - k)^2$}{b^2$} = 1$$ where $$(h, k) is $$the center. Calculate the distance $$a$$, which is the distance from the center to each vertex. Since the vertices are at (−3, 0) and (3, 0), $$a is $$the distance from the center to either vertex along the x-axis. Calculate the distance $$c$$, which is the distance from the center to each focus. The foci are at (−4, 0) and (4, 0), so $$c is $$the distance from the center to either focus along the x-axis. Use the relationship for hyperbolas: $$c^2$$ = $$a^2$$ + $$b^2. $$Solve for $$b^2$$ using the values of $$a$$ and $$c$$ found, then write the standard form equation as $$\frac{(x - h)^2$}{a^2$} - \frac{(y - k)^2$}{b^2$} = 1.$$

Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (−4, 0), (4, 0); vertices:(−3, 0), (3, 0)

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Key Concepts

Standard Form of a Hyperbola

Relationship Between Vertices, Foci, and Parameters a, b, c

Center and Orientation of the Hyperbola