Question

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

• A. 10
• B. 11
• C. 13
• D. 12

Answer 1

The Correct Option is B

Circle center coordinates:

\[ \left( \frac{3}{2}, 1 \right) \]

Diameter equation derived:

\[ 2\left( \frac{3}{2} \right) + 3(1) - k = 0 \implies k = 6 \]

Updated ellipse equation:

\[ x^2 + 9y^2 = 36 \]

\[ \frac{x^2}{6^2} + \frac{y^2}{2^2} = 1 \]

Latus rectum length calculation:

\[ LR = \frac{2b^2}{a} = \frac{2 \cdot 2^2}{6} = \frac{8}{6} = \frac{4}{3} = \frac{m}{n} \]

Final result:

\[ 2m + n = 2(4) + 3 = 11 \]