JEE Main & Advanced Mathematics Conic Sections Pole and Polar

Let \[P({{x}_{1}},{{y}_{1}})\] be any point inside or outside the ellipse. A chord through P intersects the ellipse at A and B respectively. If tangents to the ellipse at A and B meet at \[Q(h,k)\] then locus of Q is called polar of P with respect to ellipse and point P is called pole.

Equation of polar : Equation of polar of the point \[({{x}_{1}},{{y}_{1}})\] with respect to ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is given by

\[\frac{x{{x}_{1}}}{{{a}^{2}}}+\frac{y{{y}_{1}}}{{{b}^{2}}}=1\], i.e., \[T=0\].

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

Properties of pole and polar

(1) If the polar of \[P({{x}_{1}},{{y}_{1}})\]passes through \[Q({{x}_{2}},{{y}_{2}})\], 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.

(2) If the pole of a line \[{{l}_{1}}x+{{m}_{1}}y+{{n}_{1}}=0\]lies on the another line \[{{l}_{2}}x+{{m}_{2}}y+{{n}_{2}}=0\], then the pole of the second line will lie on the first and such lines are said to be conjugate lines.

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