Question

Equation of the pair of tangents drawn from the origin to the circle $ {{x}^{2}}+{{y}^{2}}+2\,g\,x+2\,fy+c=0 $ is:

• A. $ g\,x+f\,y=c({{x}^{2}}+{{y}^{2}}) $
• B. $ {{(g\,x+fy)}^{2}}={{x}^{2}}+{{y}^{2}} $
• C. $ {{(g\,x+f\,y)}^{2}}={{c}^{2}}({{x}^{2}}+{{y}^{2}}) $
• D. $ {{(g\,x+f\,y)}^{2}}=c({{x}^{2}}+{{y}^{2}}) $

Answer 1

Answer: (D)

If a tangents drawn through a point $ ({{x}_{1}},{{y}_{1}}) $ circles S, then the equation of pair of tangents is $ S{{S}_{1}}={{T}^{2}}. $ Let
$ S={{x}^{2}}+{{y}^{2}}+2\,gx+2\,fy+c=0 $
The tangents are drawn through the origin (0, 0) to the circle is
$ S{{S}_{1}}={{T}^{2}} $
$ \Rightarrow $ $ ({{x}^{2}}+{{y}^{2}}+2\,gx+2\,fy+c)c $
$ ={{(gx+fy+c)}^{2}} $
$ \Rightarrow $ $ c({{x}^{2}}+{{y}^{2}})={{(gx+fy)}^{2}} $