Question

If P and Q are the points of intersection of the circles ${x^2} + {y^2} + 3x + 7y + 2p - 5 = 0{\text{ and }}{x^2} + {y^2} + 2x + 2y - {p^2} = 0$, then there is a circle passing through P, Q and (1, 1) for
$
  {\text{A}}{\text{. all values of p}}{\text{.}} \\
  {\text{B}}{\text{. all except one value of p}}{\text{.}} \\
  {\text{C}}{\text{. all except two values of p}}{\text{.}} \\
  {\text{D}}{\text{. exactly one value of p}}{\text{.}} \\
$

Answer 1

Hint: Find the radical axis of the two circles. Then use the family of circles concept to get the circle passing through the two intersection points and the radical axis. After that, check the condition for (1, 1) to lie on the circle.

Complete step-by-step answer:
The circles given are:
$
  {S_1}:{x^2} + {y^2} + 3x + 7y + 2p - 5 = 0 \\
  {S_2}:{x^2} + {y^2} + 2x + 2y - {p^2} = 0 \\
 $
The common chord passing through the intersection of the two circles is termed as the radical axis.
So, the radical axis is determined by: ${S_1} - {S_2} = 0$
Here,
$
  {S_1} - {S_2} \\
   = {x^2} + {y^2} + 3x + 7y + 2p - 5 - ({x^2} + {y^2} + 2x + 2y - {p^2}) = 0 \\
   = x + 5y + 2p - 5 + {p^2} = 0 \\
 $
The radical axis is also in terms of p.
Now, the circle passing through the radical axis and the point of intersection of the two circles would be given by-
${S_1} + \lambda L = 0$
${S_3}:{x^2} + {y^2} + 3x + 7y + 2p - 5 + \lambda \left( {x + 5y + 2p - 5 + {p^2}} \right) = 0$ For this circle to pass through (1, 1) the point must lie on it, so substitute x=1 and y=1 in the circle equation:
$
   \Rightarrow {1^2} + {1^2} + 3\left( 1 \right) + 7\left( 1 \right) + 2p - 5 + \lambda \left( {\left( 1 \right) + 5\left( 1 \right) + 2p - 5 + {p^2}} \right) = 0 \\
   \Rightarrow 7 + 2p + \lambda \left( {1 + 2p + {p^2}} \right) = 0 \\
   \Rightarrow \lambda = \dfrac{{ - (7 + 2p)}}{{\left( {1 + 2p + {p^2}} \right)}} \\
 $
For $\lambda $ to exist, p should not get the value-
$
  1 + 2p + {p^2} \ne 0 \\
  {\left( {p + 1} \right)^2} \ne 0 \\
  p \ne - 1 \\
$
Thus p takes all values except -1.
The correct option is B.

Note: The radical axis is the common chord of the two circles which passes through their intersection points. This is why, while finding the circle passing through the two intersecting points, we took the radical axis into consideration.