Multiple choice

The equation of the circle having as a diameter, the chord $x - y - 1 = 0$ of the circle $2x^2 + 2y^2 - 2x - 6y - 25 = 0$, is

  1. $x^2 + y^2 - 3x - y - \dfrac{29}{2} = 0$
  2. $2x^2 + 2y^2 + 2x - 5y - \dfrac{29}{2} = 0$
  3. $2x^2 + 2y^2 - 6x - 2y - 21 = 0$

  4. None of these
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C Correct answer
AI explanation

To find the equation of the circle for which the intersection of the given line and circle is a diameter, we use the family of circles passing through the intersection points of S = 0 and L = 0. The required equation is given by S + lambda * L = 0. Substituting the values gives (2x^2 + 2y^2 - 2x - 6y - 25) + lambda(x - y - 1) = 0; to match the coefficient of x^2 as 2 in standard options, we set lambda = -4. This yields the equation 2x^2 + 2y^2 - 6x - 2y - 21 = 0.

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