Question:
The distance between the directrices of the ellipse
\[
\frac{x^2}{36}+\frac{y^2}{20}=1
\]
is
The distance between the directrices of the ellipse
\[
\frac{x^2}{36}+\frac{y^2}{20}=1
\]
is
Show 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
\[
x=\pm\frac{a}{e}.
\]
Updated On: Jun 24, 2026
- \(9\)
- \(6\sqrt{5}\)
- \(18\)
- \(3\sqrt{5}\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Compare with the standard equation of ellipse.
The given ellipse is \[ \frac{x^2}{36}+\frac{y^2}{20}=1 \] Comparing with \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] we get \[ a^2=36,\quad b^2=20 \] So, \[ a=6 \]
Step 2: Find the eccentricity.
For an ellipse, \[ e=\sqrt{1-\frac{b^2}{a^2}} \] Substituting the values, \[ e=\sqrt{1-\frac{20}{36}} \] \[ e=\sqrt{\frac{16}{36}} \] \[ e=\frac{2}{3} \]
Step 3: Find the directrices.
For the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] the directrices are \[ x=\pm \frac{a}{e} \] Now, \[ \frac{a}{e}=\frac{6}{\frac{2}{3}} \] \[ =6\cdot \frac{3}{2} \] \[ =9 \] Therefore, the directrices are \[ x=9 \] and \[ x=-9 \]
Step 4: Find the distance between directrices.
Distance between \[ x=9 \] and \[ x=-9 \] is \[ 9-(-9)=18 \]
Step 5: Final conclusion.
Therefore, \[ \boxed{18} \]
The given ellipse is \[ \frac{x^2}{36}+\frac{y^2}{20}=1 \] Comparing with \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] we get \[ a^2=36,\quad b^2=20 \] So, \[ a=6 \]
Step 2: Find the eccentricity.
For an ellipse, \[ e=\sqrt{1-\frac{b^2}{a^2}} \] Substituting the values, \[ e=\sqrt{1-\frac{20}{36}} \] \[ e=\sqrt{\frac{16}{36}} \] \[ e=\frac{2}{3} \]
Step 3: Find the directrices.
For the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] the directrices are \[ x=\pm \frac{a}{e} \] Now, \[ \frac{a}{e}=\frac{6}{\frac{2}{3}} \] \[ =6\cdot \frac{3}{2} \] \[ =9 \] Therefore, the directrices are \[ x=9 \] and \[ x=-9 \]
Step 4: Find the distance between directrices.
Distance between \[ x=9 \] and \[ x=-9 \] is \[ 9-(-9)=18 \]
Step 5: Final conclusion.
Therefore, \[ \boxed{18} \]
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