Correct Answer - Option 2 : No value of K
The equation of circle from the question:
S1 = x2 + y2 + 5Kx + 2y + K = 0
The equation of circle from the question:
S2 = 2(x2 + y2) + 2Kx + 3y – 1 = 0
On dividing above equation with 2,
\({{\rm{S}}_2} = {{\rm{x}}^2} + {{\rm{y}}^2} + {\rm{Kx}} + \frac{3}{2}{\rm{y}} - \frac{1}{2} = 0\)
Equation of common chord is S1 – S2 = 0
\(\Rightarrow \left( {{{\rm{x}}^2} + {{\rm{y}}^2} + 5{\rm{Kx}} + 2{\rm{y}} + {\rm{K}}} \right) - \left( {{{\rm{x}}^2} + {{\rm{y}}^2} + {\rm{Kx}} + \frac{3}{2}{\rm{y}} - \frac{1}{2}} \right) = 0\)
\(\Rightarrow {{\rm{x}}^2} + {{\rm{y}}^2} + 5{\rm{Kx}} + 2{\rm{y}} + {\rm{K}} - {{\rm{x}}^2} - {{\rm{y}}^2} - {\rm{Kx}} - \frac{3}{2}{\rm{y}} + \frac{1}{2} = 0\)
\(\Rightarrow \left( {5{\rm{K}} - {\rm{K}}} \right){\rm{x}} + \left( {2 - \frac{3}{2}} \right){\rm{y}} + {\rm{K}} + \frac{1}{2} = 0\)
\(\Rightarrow \left( {4{\rm{K}}} \right){\rm{x}} + \left( {\frac{{4 - 3}}{2}} \right){\rm{y}} + {\rm{K}} + \frac{1}{2} = 0\)
\(\Rightarrow \left( {4{\rm{K}}} \right){\rm{x}} + \left( {\frac{1}{2}} \right){\rm{y}} + {\rm{K}} + \frac{1}{2} = 0\)
\(\Rightarrow 4{\rm{Kx}} + \frac{{\rm{y}}}{2} + {\rm{K}} + \frac{1}{2} = 0\) ----(1)
Equation of the line passing through the intersection points P & Q is,
4x + 5y – K = 0 ----(2)
On comparing equation (1) and (2),
\(\frac{{4{\rm{K}}}}{4} = \frac{1}{{10}} = \frac{{2{\rm{K}} + 1}}{{ - 2{\rm{K}}}}\) ----(3)
\(\Rightarrow \frac{{4{\rm{K}}}}{4} = \frac{1}{{10}}\)
\(\therefore {\rm{K}} = \frac{1}{{10}}\)
\(\Rightarrow \frac{1}{{10}} = \frac{{2{\rm{K}} + 1}}{{ - 2{\rm{K}}}}\)
-2K = 20K + 10
⇒ 22K = -10
\(K = - \frac{{10}}{{22}}\)
\(\therefore {\rm{K}} = \frac{{ - 5}}{{11}}\)
\({\rm{K}} = \frac{1}{{10}}{\rm{\;or\;}}\frac{{ - 5}}{{11}}\) is not satisfying equation (3)
Therefore, no value of K exists.
