JEE Main & Advanced Mathematics Conic Sections Equation of Pair of Tangents

If \[P({{x}_{1}},\,{{y}_{1}})\] be any point outside the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] then a pair of tangents PQ, PR can be drawn to it from P.

The equation of pair of tangents PQ and PR is \[S{{S}_{1}}={{T}^{2}}\]   

where,\[S=\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}-1\],\[{{S}_{1}}=\frac{x_{1}^{2}}{{{a}^{2}}}-\frac{y_{1}^{2}}{{{b}^{2}}}-1,\,T=\frac{x{{x}_{1}}}{{{a}^{2}}}-\frac{y{{y}_{1}}}{{{b}^{2}}}-1\]

Director circle : The director circle is the locus of points from which perpendicular tangents are drawn to the given hyperbola. The equation of the director circle of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}-{{b}^{2}}\].