Question:
If the latus rectum of an ellipse is equal to half of the minor axis, then its eccentricity is:
If the latus rectum of an ellipse is equal to half of the minor axis, then its eccentricity is:
Show Hint
For an ellipse,
\[
\text{Length of latus rectum}=\frac{2b^2}{a}
\]
and
\[
e=\sqrt{1-\frac{b^2}{a^2}}
\]
These two standard formulas are frequently used together in problems involving eccentricity.
Updated On: Jun 25, 2026
- \(\dfrac{\sqrt{3}}{4}\)
- \(\dfrac{3}{4}\)
- \(\dfrac{1}{4}\)
- \(\dfrac{\sqrt{3}}{2}\)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Formula for the length of latus rectum of an ellipse.
For the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] the length of the latus rectum is \[ \frac{2b^2}{a} \] The length of the minor axis is \[ 2b \] According to the question, \[ \frac{2b^2}{a}=\frac{1}{2}(2b) \]
Step 2: Simplify the equation.
We get \[ \frac{2b^2}{a}=b \] Multiplying both sides by \(a\), \[ 2b^2=ab \] Dividing by \(b\neq 0\), \[ 2b=a \] Hence, \[ b=\frac{a}{2} \]
Step 3: Use the eccentricity formula.
For an ellipse, \[ e=\sqrt{1-\frac{b^2}{a^2}} \] Substituting \[ b=\frac{a}{2}, \] we get \[ e=\sqrt{1-\frac{(a/2)^2}{a^2}} \] \[ =\sqrt{1-\frac{1}{4}} \] \[ =\sqrt{\frac{3}{4}} \] \[ =\frac{\sqrt{3}}{2} \]
Step 4: Final conclusion.
Therefore, the eccentricity of the ellipse is \[ \boxed{\frac{\sqrt{3}}{2}} \]
For the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] the length of the latus rectum is \[ \frac{2b^2}{a} \] The length of the minor axis is \[ 2b \] According to the question, \[ \frac{2b^2}{a}=\frac{1}{2}(2b) \]
Step 2: Simplify the equation.
We get \[ \frac{2b^2}{a}=b \] Multiplying both sides by \(a\), \[ 2b^2=ab \] Dividing by \(b\neq 0\), \[ 2b=a \] Hence, \[ b=\frac{a}{2} \]
Step 3: Use the eccentricity formula.
For an ellipse, \[ e=\sqrt{1-\frac{b^2}{a^2}} \] Substituting \[ b=\frac{a}{2}, \] we get \[ e=\sqrt{1-\frac{(a/2)^2}{a^2}} \] \[ =\sqrt{1-\frac{1}{4}} \] \[ =\sqrt{\frac{3}{4}} \] \[ =\frac{\sqrt{3}}{2} \]
Step 4: Final conclusion.
Therefore, the eccentricity of the ellipse is \[ \boxed{\frac{\sqrt{3}}{2}} \]
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