A) exactly one value of a
B) no value of a
C) infinitely many values of a
D) exactly two values of a
Correct Answer: B
Solution :
| \[{{S}_{1}}={{x}^{2}}+{{y}^{2}}+2ax+cy+a=0\] |
| \[{{S}_{2}}={{x}^{2}}+{{y}^{2}}-3ax+dy-1=0\] |
| Equation of common chord of circles \[{{S}_{1}}\] and \[{{S}_{2}}\] is given by \[{{S}_{1}}-{{S}_{2}}=0\] |
| \[\Rightarrow \,\,5ax+(c-d)y+a+1=0\] |
| Given that \[5x+by-a=0\] passes through P and Q |
| \[\therefore \] The two equations should represent the same line |
| \[\Rightarrow \,\,\,\frac{a}{1}=\frac{c-d}{b}=\frac{a+1}{-a}\Rightarrow a+1=-{{a}^{2}}\] |
| \[\Rightarrow \,\,{{a}^{2}}+a+1=0\] |
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