Question

The distance between the directrices of the ellipse \[ \frac{x^2}{36}+\frac{y^2}{20}=1 \] is

• A. \(9\)
• B. \(6\sqrt{5}\)
• C. \(18\)
• D. \(3\sqrt{5}\)

Hint:

For an ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] where \(a\gt b\), eccentricity is \[ e=\sqrt{1-\frac{b^2}{a^2}}, \] and the directrices are \[ x=\pm\frac{a}{e}. \]

Answer 1

The Correct Option is C

Step 1: Read off $a^2$ and $b^2$.
From $\dfrac{x^2}{36} + \dfrac{y^2}{20} = 1$, we get $a^2 = 36$, $b^2 = 20$, so $a = 6$. Since $a > b$, the major axis is along the x-axis.

Step 2: Find eccentricity $e$.
For an ellipse, $b^2 = a^2(1 - e^2)$: \[ 20 = 36(1 - e^2) \Rightarrow 1 - e^2 = \frac{20}{36} = \frac{5}{9} \Rightarrow e^2 = \frac{4}{9} \Rightarrow e = \frac{2}{3}. \]

Step 3: Recall the directrix formula.
For an ellipse with major axis along x, the two directrices are at $x = \pm \dfrac{a}{e}$.

Step 4: Calculate $a/e$.
\[ \frac{a}{e} = \frac{6}{2/3} = 6 \times \frac{3}{2} = 9. \] So the directrices are at $x = 9$ and $x = -9$.

Step 5: Find the distance between the directrices.
\[ \text{Distance} = 9 - (-9) = 18. \]

Step 6: State the answer.
The distance between the directrices is 18.
\[ \boxed{18} \]