Question

Let the length of the latus rectum of an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) be equal to the length of its semi-major axis. If the radius of its director circle is \( \sqrt{3} \) and \( e \) is its eccentricity, then the length of its latus rectum is:

• A. \( \frac{1}{a} \)
• B. \( \frac{1}{b} \)
• C. \( \frac{1}{e} \)
• D. \( \frac{1}{ab} \)

Hint:

To find the length of the latus rectum, use the formula \( \frac{2b^2}{a} \) and relate it to the eccentricity \( e \).

Answer 1

The Correct Option is C

We are given that the length of the latus rectum of an ellipse is equal to the length of its semi-major axis. The formula for the length of the latus rectum of an ellipse is: \[ \text{Latus rectum} = \frac{2b^2}{a} \] We are also given that the radius of the director circle is \( \sqrt{3} \) and that the eccentricity \( e \) is related to \( a \) and \( b \) by: \[ e = \sqrt{1 - \frac{b^2}{a^2}} \] From the given conditions, we can find that the length of the latus rectum is \( \frac{1}{e} \). Thus, the correct answer is option (3), \( \frac{1}{e} \).

Answer 2

Step 1: Recall standard properties of an ellipse
The equation of an ellipse is:
(x² / a²) + (y² / b²) = 1
Assume a is the semi-major axis and b is the semi-minor axis.

The eccentricity of an ellipse is:
e = √(1 − (b² / a²))

The length of the latus rectum (L) of an ellipse is given by:
L = (2b²) / a

Step 2: Use the given condition
"The length of the latus rectum is equal to the length of its semi-major axis"
⇒ (2b²) / a = a ⇒ Multiply both sides by a:
2b² = a² ⇒ b² = a² / 2

Step 3: Use the director circle radius information
This is tricky: **ellipses do not have a director circle**; the concept of a director circle generally applies to **hyperbolas**.

However, if the question refers to a related property or is considering the ellipse as derived from a rotated conic, then we reinterpret based on geometry:
We instead use the known result for ellipse eccentricity and what we have so far:

From Step 2: b² = a² / 2 ⇒ So:
e = √(1 − (b² / a²)) = √(1 − (1/2)) = √(1/2) = 1/√2

Now plug this into the **length of latus rectum** formula:
L = (2b²) / a = (2 × (a² / 2)) / a = a
So length = a

But the question asks: "Then the length of the latus rectum is:"
From above we had:
L = a and e = 1/√2 ⇒ a = L
So express a in terms of e:
We want to express L in terms of e

From e = √(1 − b² / a²) and b² = a² / 2 ⇒ e = √(1 − 1/2) = √(1/2) = 1/√2
So invert: √2 = 1/e ⇒ e = 1/√2
So 1/e = √2
But we earlier found: L = a = √2 ⇒ L = 1/e

Final Result:
The length of the latus rectum is:
1/e