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 and the length of the latus rectum of the ellipse is , where m and n are

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Solution

Q.
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 $x^{2} + y^{2} - 3 x - 2 y = 0$ and the length of the latus rectum of the ellipse $x^{2} + 9 y^{2} = k^{2}$ is $\frac{m}{n}$ , where m and n are coprime, then 2m + n is equal to [2024]

1 10

2 13

3 12

4 11

Ans.
(4)

Equation of circle with AB as diameter is

$(x - \frac{k}{2}) x + y (y - \frac{k}{3}) = 0$

$\Rightarrow x^{2} + y^{2} - \frac{k x}{2} - \frac{k y}{3} = 0$

On comparing it with given equation $x^{2} + y^{2} - 3 x - 2 y = 0$

we get $\frac{k}{2} = 3 \Rightarrow k = 6$

Now, equation of ellipse is given by

$x^{2} + 9 y^{2} = 36$

$\Rightarrow \frac{x^{2}}{36} + \frac{y^{2}}{4} = 1$

So latus rectum $\frac{2 b^{2}}{a} = \frac{2 \times 4}{6} = \frac{4}{3} = \frac{m}{n}$

$\Rightarrow 2 m + n = 2 \times 4 + 3 = 11$ .

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