Let the line 2x + 3y – k = 0, k > 0, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is x2 + y2 – 3x – 2y = 0 and the length of the latus rectum of the ellipse x2 + 9y2 = k2 is m n , where m and n are coprime, then 2m + n is equal to
- 10
- 11
- 13
- 12
The Correct Option is B
Solution and Explanation
Centre of the circle:
\[ \left( \frac{3}{2}, 1 \right) \]
Equation of diameter:
\[ 2\left( \frac{3}{2} \right) + 3(1) - k = 0 \implies k = 6 \]
Now, equation of ellipse becomes:
\[ x^2 + 9y^2 = 36 \]
\[ \frac{x^2}{6^2} + \frac{y^2}{2^2} = 1 \]
Length of latus rectum (LR):
\[ LR = \frac{2b^2}{a} = \frac{2 \cdot 2^2}{6} = \frac{8}{6} = \frac{4}{3} = \frac{m}{n} \]
Thus, \[ 2m + n = 2(4) + 3 = 11 \]
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