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JEE Main 2005
Exam
JEE Main
Year
2005
Subject
Mathematics
Chapter
Circle

JEE Main 2005 · Mathematics · Circle

If the circles x^2+y^2+2 a x+c y+a=0 and x^2+y^2-3 a x+d y-1=0...

If the circles x2+y2+2ax+cy+a=0x^2+y^2+2 a x+c y+a=0 and x2+y2−3ax+dy−1=0x^2+y^2-3 a x+d y-1=0 intersect in two distinct points PP and QQ then the line 5x+5 x+ by −a=0-a=0 passes through PP and QQ for

Options

  1. Aexactly one value of a
  2. Bno value of a
  3. Cinfinitely many values of a
  4. Dexactly two values of a

Correct answer

B. no value of a

Step-by-step solution

S1=x2+y2+2ax+cy+a=0S2=x2+y2−3ax+dy−1=0\begin{aligned}& S_1=x²+y²+2 a x+c y+a=0 \\ & S_2=x²+y²-3 a x+d y-1=0 \end{aligned}EquationofradicalaxisofS1andS2of radical axis ofS_1 and S_2S1−S2=0⇒5ax+(c−d)y+a+1=0\begin{aligned}& S_1-S_2=0 \\ & \Rightarrow 5 a x+(c-d) y+a+1=0 \end{aligned}GiventhatGiven that5 \mathrm{x}+\mathrm{by}-\mathrm{a}=0passesthroughpasses through\mathrm{P}andand\mathrm{Q} ⇒a1=c−db=a+1−a⇒a+1=−a2a2+a+1=0\begin{aligned}& \Rightarrow \frac{a}{1}=\frac{c-d}{b}=\frac{a+1}{-a}\\ & \Rightarrow a+1=-a² \\ & a²+a+1=0 \end{aligned} No real value of a.
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