Question

Assertion (A): The length of the latus rectum of an ellipse is 4. The focus and its corresponding directrix are respectively \( (1, -2) \) and the line \( 3x + 4y - 15 = 0 \). Then its eccentricity is \( \dfrac{1}{2} \). Reason (R): Length of the perpendicular drawn from focus of an ellipse to its corresponding directrix is \( \dfrac{a(1 - e^2)}{e} \)
Then which one of the following is correct?

• (A) and (R) are true, and (R) is the correct explanation to (A)
• (A) and (R) are true, and (R) is not the correct explanation to (A)
• (A) is true, (R) is false
• (A) is false, (R) is true

Hint:

For assertion-reason type questions in conic sections, always verify the mathematical relationship behind the reason and check whether it directly explains the assertion.

Answer 1

The Correct Option is A

Step 1: Use the formula for latus rectum. For an ellipse, the length of the latus rectum is \( \dfrac{2b^2}{a} \), and eccentricity is \( e = \dfrac{c}{a} \). We're told the latus rectum is 4 and \( e = \frac{1}{2} \), so: Let \( a \) be semi-major axis. Since \( e = \frac{1}{2} \), we know that: \[ b^2 = a^2(1 - e^2) = a^2 \left(1 - \frac{1}{4}\right) = \frac{3a^2}{4} \] \[ \text{Latus rectum} = \frac{2b^2}{a} = \frac{2 . \frac{3a^2}{4}}{a} = \frac{3a}{2} \] Set this equal to 4: \[ \frac{3a}{2} = 4 \Rightarrow a = \frac{8}{3} \] Step 2: Validate the perpendicular formula (R). The perpendicular from the focus to its corresponding directrix is given by: \[ \text{Distance} = \frac{a(1 - e^2)}{e} \] This matches the given Reason (R), which is correct. Step 3: Explanation correctness. Since the directrix and focus are given, we can compute the perpendicular distance and verify the eccentricity as \( \frac{1}{2} \). Thus, both statements are true and (R) correctly explains (A).