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

Answer: (B)

11

Answer 2

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, $2m + n = 2(4) + 3 = 11$