JEE Main & Advanced Mathematics Conic Sections Pole and Polar

The locus of the point of intersection of the tangents to the hyperbola at A and B is called the polar of the given point \[P\] with respect to the hyperbola and the point \[P\] is called the pole of the polar. The equation of the required polar with \[({{x}_{1}},\,{{y}_{1}})\] as its pole is  \[\frac{x{{x}_{1}}}{{{a}^{2}}}-\frac{y{{y}_{1}}}{{{b}^{2}}}=1\].  

Pole of a given line: The pole of a given line \[lx+my+n=0\] with respect to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[({{x}_{1}},{{y}_{1}})=\]\[\left( -\frac{{{a}^{2}}l}{n},\,\frac{{{b}^{2}}m}{n} \right)\].

Properties of pole and polar

(i) If the polar of \[P({{x}_{1}},\,{{y}_{1}})\] passes through \[{{y}_{1}},{{y}_{2}},\,{{y}_{3}}\], then the polar of \[Q({{x}_{2}},{{y}_{2}})\] goes through \[P({{x}_{1}},\,{{y}_{1}})\] and such points are said to be conjugate points.

(ii) If the pole of a line \[lx+my+n=0\] lies on the another line \[4{{x}^{2}}-(4h-k)\,x-1=0\] then the pole of the second line will lie on the first and such lines are said to be conjugate lines.

(iii) Pole of a given line is same as point of intersection of tangents as its extremities.