Question

The latus rectum of the hyperbola $3x^{2}-2y^{2}=6$ is

• A. $\frac{3}{\sqrt{2}}$
• B. $\frac{4}{\sqrt{3}}$
• C. $\frac{2}{\sqrt{3}}$
• D. 3
• E. $3\sqrt{2}$

Hint:

Formula Tip: The length of the latus rectum for an ellipse and a hyperbola is exactly the same: $\frac{2b^2}{a}$ (assuming $a$ is the semi-major/transverse axis).

Answer 1

Answer: (E)

Concept:
To find the length of the latus rectum of a hyperbola, first convert its equation to the standard form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. The formula for the length of the latus rectum of a standard horizontal hyperbola is $\text{L.R.} = \frac{2b^2}{a}$.

Step 1: State the given equation.
The given equation of the hyperbola is: $$3x^2 - 2y^2 = 6$$

Step 2: Convert the equation to standard form.
Divide the entire equation by 6 to make the right side equal to 1: $$\frac{3x^2}{6} - \frac{2y^2}{6} = \frac{6}{6}$$ $$\frac{x^2}{2} - \frac{y^2}{3} = 1$$

Step 3: Identify the parameters $a$ and $b^2$.
By comparing the equation to $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, we extract: $$a^2 = 2 \implies a = \sqrt{2}$$ $$b^2 = 3$$

Step 4: Set up the latus rectum formula.
Use the formula for the length of the latus rectum: $$\text{L.R.} = \frac{2b^2}{a}$$

Step 5: Substitute and calculate the final length.
Plug in $b^2 = 3$ and $a = \sqrt{2}$: $$\text{L.R.} = \frac{2(3)}{\sqrt{2}}$$ $$\text{L.R.} = \frac{6}{\sqrt{2}}$$ To rationalize the denominator, multiply the top and bottom by $\sqrt{2}$: $$\text{L.R.} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}$$ Hence the correct answer is (E) $3\sqrt{2$}.