Question:
If the ellipse centered at the origin has a focus at \((0,3)\) and the corresponding directrix is \(y=9\), then which one of the following is not true for this ellipse ?
If the ellipse centered at the origin has a focus at \((0,3)\) and the corresponding directrix is \(y=9\), then which one of the following is not true for this ellipse ?
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Use \(ae=3\) and \(\frac{a}{e}=9\) to get \(a^{2}\) and \(b^{2}\), then test each statement.
Updated On: Nov 12, 2026
- Length of its minor axis is \(6\sqrt{2}\)
- Length of its latus rectum is \(4\sqrt{3}\)
- Point \((\sqrt{6},\,2\sqrt{3})\) lies on it
- Point \((\sqrt{6},\,-3\sqrt{2})\) lies on it
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The Correct Option is C
Solution and Explanation
The focus is on the y-axis, so the major axis is vertical: \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\).
Focus at distance \(ae=3\) and directrix at distance \(\frac{a}{e}=9\). Multiplying, \(a^{2}=27\); then \(e=\frac{3}{a}=\frac{1}{\sqrt{3}}\).
\(b^{2}=a^{2}(1-e^{2})=27\cdot\frac{2}{3}=18\). Ellipse: \(\frac{x^{2}}{18}+\frac{y^{2}}{27}=1\).
Minor axis \(=2b=2\cdot3\sqrt{2}=6\sqrt{2}\): true.
Latus rectum \(=\frac{2b^{2}}{a}=\frac{36}{3\sqrt{3}}=4\sqrt{3}\): true.
Point \((\sqrt{6},2\sqrt{3})\): \(\frac{6}{18}+\frac{12}{27}=\frac{1}{3}+\frac{4}{9}=\frac{7}{9}\ne 1\): does not lie on it, so the statement is not true.
Point \((\sqrt{6},-3\sqrt{2})\): \(\frac{6}{18}+\frac{18}{27}=\frac{1}{3}+\frac{2}{3}=1\): lies on it, true.
Answer: (3) Point \((\sqrt{6},\,2\sqrt{3})\) lies on it
Focus at distance \(ae=3\) and directrix at distance \(\frac{a}{e}=9\). Multiplying, \(a^{2}=27\); then \(e=\frac{3}{a}=\frac{1}{\sqrt{3}}\).
\(b^{2}=a^{2}(1-e^{2})=27\cdot\frac{2}{3}=18\). Ellipse: \(\frac{x^{2}}{18}+\frac{y^{2}}{27}=1\).
Minor axis \(=2b=2\cdot3\sqrt{2}=6\sqrt{2}\): true.
Latus rectum \(=\frac{2b^{2}}{a}=\frac{36}{3\sqrt{3}}=4\sqrt{3}\): true.
Point \((\sqrt{6},2\sqrt{3})\): \(\frac{6}{18}+\frac{12}{27}=\frac{1}{3}+\frac{4}{9}=\frac{7}{9}\ne 1\): does not lie on it, so the statement is not true.
Point \((\sqrt{6},-3\sqrt{2})\): \(\frac{6}{18}+\frac{18}{27}=\frac{1}{3}+\frac{2}{3}=1\): lies on it, true.
Answer: (3) Point \((\sqrt{6},\,2\sqrt{3})\) lies on it
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