Question

Find an equation in standard form for the hyperbola with vertices at (0, ±2) and foci at (0, ±7).

Answer 1

Answer on Question #45115 – Math – Analytic Geometry

Question

Find an equation in standard form for the hyperbola with vertices at $(0; \pm 2)$ and foci at $(0; \pm 7)$.

Solution

Since the vertices and foci are located on the $y$-axis, the general equation of this hyperbola has the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$. Find $a^2$ and $b^2$. Let $x = 0, y = \pm 2$. Then we have

$$- \frac {4}{b ^ {2}} = - 1 \Leftrightarrow b ^ {2} = 4.$$

By hypothesis, a half of focal length $c$ is equal to 7, so we have

$$c ^ {2} = a ^ {2} + b ^ {2} \Rightarrow 49 = a ^ {2} + 4 \Rightarrow a ^ {2} = 45.$$

Answer: $\frac{x^2}{45} - \frac{y^2}{4} = -1$.

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