Question

The eccentricity of the hyperbola which passes through the points $(3, 0)$ and $(3\sqrt{2}, 2)$ is \dots

• A. $\sqrt{13}$
• B. $\sqrt{13}/4$
• C. $\sqrt{13}/3$
• D. $\sqrt{13}/2$

Hint:

When a conic section passes through a point of the form $(x_0, 0)$, it immediately reveals the length of the semi-major (or transverse) axis, because the $y$-term zeroes out entirely.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The standard equation of a hyperbola is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. The eccentricity is $e = \sqrt{1 + \frac{b^2}{a^2}}$.

Step 2: Formula Application:
Substitute the given points into the equation to find $a^2$ and $b^2$.

Step 3: Explanation:
Point $(3, 0): \frac{9}{a^2} - 0 = 1 \implies a^2 = 9$. Point $(3\sqrt{2}, 2): \frac{18}{9} - \frac{4}{b^2} = 1 \implies 2 - 1 = \frac{4}{b^2} \implies b^2 = 4$. $e = \sqrt{1 + \frac{4}{9}} = \sqrt{\frac{13}{9}} = \frac{\sqrt{13}}{3}$.

Step 4: Final Answer:
The eccentricity is $\sqrt{13}/3$.