Question

The equation of the hyperbola with vertices \( (0, \pm 15) \) and foci \( (0, \pm 20) \) is:

• A. \( \frac{x^2}{175} - \frac{y^2}{225} = 1 \)
• B. \( \frac{x^2}{625} - \frac{y^2}{125} = 1 \)
• C. \( \frac{y^2}{225} - \frac{x^2}{125} = 1 \)
• D. \( \frac{y^2}{65} - \frac{x^2}{65} = 1 \)
• E. \( \frac{y^2}{225} - \frac{x^2}{175} = 1 \)

Hint:

Always check which coordinate is changing in the vertices. If the y-coordinate changes (like here), the \( y^2 \) term is positive. If the x-coordinate changes, the \( x^2 \) term is positive.

Answer 1

Answer: (E)

Concept: Since the vertices and foci lie on the y-axis (x-coordinates are zero), this is a vertical hyperbola. Its standard equation is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). In this form, \( a \) is the distance from the center to the vertices, and \( c \) is the distance from the center to the foci. The relationship between these values is \( c^2 = a^2 + b^2 \).

Step 1: Identify the values of \( a \) and \( c \).
The vertices are \( (0, \pm 15) \), so the distance \( a = 15 \). Therefore, \( a^2 = 15^2 = 225 \). The foci are \( (0, \pm 20) \), so the distance \( c = 20 \). Therefore, \( c^2 = 20^2 = 400 \).

Step 2: Calculate \( b^2 \) and form the equation.
Using the relationship \( c^2 = a^2 + b^2 \): \[ 400 = 225 + b^2 \] \[ b^2 = 400 - 225 = 175 \] Now, substitute \( a^2 = 225 \) and \( b^2 = 175 \) into the vertical hyperbola formula: \[ \frac{y^2}{225} - \frac{x^2}{175} = 1 \]