A) only one value of a
B) no value of a
C) infinite values of a
D) only two values of a
Correct Answer: B
Solution :
Given equations of the circles are \[{{S}_{1}}={{x}^{2}}+{{y}^{2}}+2ax+cy+a=0\] and \[{{S}_{2}}={{x}^{2}}+{{y}^{2}}-3ax+dy-1=0\] The equation of chord which passes through the intersection points of\[{{S}_{1}}\equiv 0\]and\[{{S}_{2}}\equiv 0\]is \[{{S}_{1}}-{{S}_{2}}=0\] ie, \[5ax+(c-d)y+a+1=0\] On comparing with\[5x+by-a=0,\] we get \[\frac{5a}{5}=\frac{c-d}{b}=\frac{a+1}{-a}\] \[\Rightarrow \] \[a(-a)=a+1\] \[\Rightarrow \] \[{{a}^{2}}+a+1=0\] So, this line will not passes through P and Q for any value of a.
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