Given the hyperbola $\frac{y^2}{100}-\frac{x^2}{139}=1$, find the length of the $a$-axis and the $b$-axis.

• A. $a=100,b=139$
• B. $a=139,b=100$
• C. $a=\sqrt{139},b=10$
• D. $a=10,b=\sqrt{139}$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize the standard form of the hyperbola equation. The given equation is \( \frac{y^2}{100} - \frac{x^2}{139} = 1 \), which represents a vertical hyperbola because the \( y^2 \) term is positive and comes first.
2. Step 2: Identify the values of \( a^2 \) and \( b^2 \) from the denominators of the equation. Here, \( a^2 = 100 \) and \( b^2 = 139 \).
3. Step 3: Calculate \( a \) and \( b \) by taking the square root of \( a^2 \) and \( b^2 \). This gives \( a = \sqrt{100} = 10 \) and \( b = \sqrt{139} \).
4. Step 4: Understand the geometric significance of \( a \) and \( b \). For a vertical hyperbola, \( a \) represents the distance from the center to the vertices along the \( y \)-axis, and \( b \) represents the distance from the center to the co-vertices along the \( x \)-axis.
5. Step 5: The lengths of the \( a \)-axis and \( b \)-axis are determined as follows: The \( a \)-axis length is \( 2a \), which equals \( 2 \times 10 = 20 \), and the \( b \)-axis length is \( 2b \), which equals \( 2 \times \sqrt{139} \).