AP EAMCET · Maths · Ellipse
Assertion (A) : The length of the latus rectum of an ellipse is 4. The focus and its corresponding directrix are respectively \((1,-2)\) and \(3 x+4 y-15=0\). Then its eccentricity is \(\frac{1}{2}\).
Reason (R) : Length of the perpendicular drawn from focus of an ellipse to its corresponding directrix is \(\frac{a\left(1-e^2\right)}{e}\).
Then which one of the following is correct ?
- A (A) and (R) are true, and (R) is the correct explanation to (A)
- B (A) and (R) are true, and (R) is not the correct explanation to (A)
- C (A) is true, (R) is false
- D (A) is false, (R) is true
Answer & Solution
Correct Answer
(A) (A) and (R) are true, and (R) is the correct explanation to (A)
Step-by-step Solution
Detailed explanation
Assertion (A): Distance from focus to directrix \(d = \frac{|3(1) + 4(-2) - 15|}{\sqrt{3^2 + 4^2}} = \frac{|3 - 8 - 15|}{\sqrt{9 + 16}} = \frac{|-20|}{5} = 4\). Length of latus rectum \(L = 2b^2/a = 4 \implies b^2/a = 2\). Also, \(b^2 = a^2(1-e^2)\), so…
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