• # question_answer Let the length of the tangent drawn from a variable point to one. given circle is$\mu \,(\mu \ne 1)$times the length of the tangent from it to another circle. The locus of the variable point is A)  a circle                        B)  a parabola         C)  a straight line              D)  a hyperbola

We have, $P{{T}_{1}}=\mu P{{T}_{2}}$    (by condition) $\Rightarrow$            ${{\alpha }^{2}}+{{\beta }^{2}}+2{{g}_{1}}\alpha +2{{f}_{1}}\beta +{{c}_{1}}$ $={{\mu }^{2}}\,({{\alpha }^{2}}+{{\beta }^{2}}+2{{g}_{2}}\alpha +2{{f}_{2}}\beta +{{c}_{2}})$ Hence, locus of $(\alpha ,\,\,\beta )$ is $({{x}^{2}}+{{y}^{2}})\,({{\mu }^{2}}-1)+2({{g}_{2}}{{\mu }^{2}}-{{g}_{1}})x$ $+2\,({{f}_{2}}{{\mu }^{2}}-{{f}_{1}})\,y+{{c}_{2}}{{\mu }^{2}}-{{c}_{1}}=0$ Which is a circle.