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JEE Main20245 Apr 2024Morning ShiftMathematicsEllipseActual

Let the line 2 x+3 y- 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 m n , where m and n are coprime, then 2 ~m + n is equal to

Options

  1. A11
  2. B10
  3. C12
  4. D13

Correct answer

A. 11

Step-by-step solution

Centre of the circle = ( 3 2 , 1 ) Equation of diameter =2 x +3 y - k =0 aligned & 2 ( 3 2 )+3(1)- k =0 & k =6 aligned Now, Equation of ellipse becomes aligned & x^2+9 y^2=36 & x^2 6^2 + y^2 2^2 =1 aligned aligned & length of L R= 2 b^2 a = 2.2^2 6 = 8 6 = 4 3 = m n & 2 m+n=2(4)+3=11 aligned

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