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KCET2010MathematicsCircle

The straight line 2 x+3 y-k=0, k>0 cuts the x and y -axes at A and B . The area of OAB , where O is the origin, is 12 sq unit. The equation of the circle having A B as diameter is

Options

  1. Ax²+y²-6 x-4 y=0
  2. Bx²+y²+4 x-6 y=0
  3. Cx²+y²-6 x+4 y=0
  4. Dx²+y²-4 x-6 y=0

Correct answer

A. x²+y²-6 x-4 y=0

Step-by-step solution

Given, equation of line, 2 x+3 y-k=0, k>0 . x k 2 + y k 3 =1 ...(i) Given area of OAB =12 gathered = 1 2 | array ccc 0 & 0 & 1 k 2 & 0 & 1 0 & k 3 & 1 array |=12 k ² 6 =24 k ²=144 k =12 gathered Hence, the coordinate of A(6,0) and B(0,4) . Then, diameter = AB aligned &= (6-0)²+(0-4)² &= 36+16 &= 52 aligned Centre = Mid point of A B = ( 6+0 2 , 0+4 2 )=(3,2) The equation of circle is (x-3)²+(y-2)²= ( 52 2 )² x ²+9-6 x + y ²+4-4 y =13 x²+y²-6 x-4 y+13=13 x²+y²-6 x-4 y=0

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