Correct Answer - Option 4 : (x + 2)
2 + y
2 = 16
From the given question, potential for q and -2q
\({V_q} = \frac{q}{{4\pi\epsilon \sqrt {{x^2} + {y^2}} }}\)
\({V_{ - 2q}} = \frac{{ - 2q}}{{4\pi\epsilon \left( {\sqrt {{{\left( {x - 6} \right)}^2} + {y^2}} } \right)}}\)
For equipotential region in x-y plane,
\({V_{total}} = 0 = \frac{q}{{4\pi\epsilon \left( {\sqrt {{x^2} + {y^2}} } \right)}} + \frac{{ - 2q}}{{4\pi\epsilon \left( {\sqrt {{{\left( {x - 6} \right)}^2} + {y^2}} } \right)}}\)
\(\begin{array}{l} \sqrt {{{\left( {x - 6} \right)}^2} + {y^2}} = 2\left( {\sqrt {{x^2} + {y^2}} } \right)\\ 3{x^2}\; + \;3{y^2}\; + 12x\; = \;36\\ {x^2}\; + \;{y^2}\; + \;4x\; = \;12\\ {\left( {x\; + \;2} \right)^2}\; + \;{y^2}\; = \;16 \end{array}\)