Question:
X axis is the major axis and origin is the centre of an ellipse. If the distance between its directrices is \(\frac{18}{\sqrt{5}}\) and the ratio between the distances from the centre of this ellipse to its focus and its corresponding directrices is \(5:9\), then the length of its latus rectum is
X axis is the major axis and origin is the centre of an ellipse. If the distance between its directrices is \(\frac{18}{\sqrt{5}}\) and the ratio between the distances from the centre of this ellipse to its focus and its corresponding directrices is \(5:9\), then the length of its latus rectum is
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Always verify if the ratio provided is \(ae:a/e\) or \(a/e:ae\) before computing eccentricity.
Updated On: Jun 9, 2026
- 8/5
- 9/5
- 8/3
- 16/3
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The Correct Option is C
Solution and Explanation
Concept:
For an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\):
1) Distance between directrices = \(2a/e = 18/\sqrt{5}\).
2) Distance from center to focus = \(ae\).
3) Distance from center to directrix = \(a/e\).
Ratio \(ae / (a/e) = e^2 = 5/9\).
Step 1: Determine eccentricity \(e\).
Given \(e^2 = 5/9\), so \(e = \sqrt{5}/3\).
Step 2: Find \(a\) using the distance between directrices.
\[ \frac{2a}{e} = \frac{18}{\sqrt{5}} \implies \frac{2a}{\sqrt{5}/3} = \frac{18}{\sqrt{5}} \] \[ \frac{6a}{\sqrt{5}} = \frac{18}{\sqrt{5}} \implies 6a = 18 \implies a = 3 \]
Step 3: Calculate the length of the latus rectum.
Latus rectum \(LR = \frac{2b^2}{a}\). We know \(b^2 = a^2(1 - e^2)\). \[ b^2 = 3^2 (1 - 5/9) = 9(4/9) = 4 \] \[ LR = \frac{2 \cdot 4}{3} = 8/3 \] center minipage0.3
LR = 8/3 minipage center
Step 1: Determine eccentricity \(e\).
Given \(e^2 = 5/9\), so \(e = \sqrt{5}/3\).
Step 2: Find \(a\) using the distance between directrices.
\[ \frac{2a}{e} = \frac{18}{\sqrt{5}} \implies \frac{2a}{\sqrt{5}/3} = \frac{18}{\sqrt{5}} \] \[ \frac{6a}{\sqrt{5}} = \frac{18}{\sqrt{5}} \implies 6a = 18 \implies a = 3 \]
Step 3: Calculate the length of the latus rectum.
Latus rectum \(LR = \frac{2b^2}{a}\). We know \(b^2 = a^2(1 - e^2)\). \[ b^2 = 3^2 (1 - 5/9) = 9(4/9) = 4 \] \[ LR = \frac{2 \cdot 4}{3} = 8/3 \] center minipage0.3
LR = 8/3 minipage center
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