Multiple choice

The value of $c$ for which the quadrilateral formed by the common chord of the circles $x^{2}+y^{2}+2x+2y=0$ and $x^{2}+y^{2}+6x+4y+c=0$ and the line segment joining their centre as diagonals is a rhombus is

  1. $11$
  2. $7$
  3. $9$
  4. $2$
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A Correct answer
Explanation

The common chord equation is (x^2+y^2+6x+4y+c) - (x^2+y^2+2x+2y) = 0, which is 4x + 2y + c = 0. For the quadrilateral to be a rhombus, the diagonals must bisect each other at 90 degrees. This condition leads to c = 11.

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