JEE Main & Advanced Mathematics Circle and System of Circles Pole and Polar

Let \[P\,({{x}_{1}},\,{{y}_{1}})\] be any point inside or outside the circle. Draw chords AB and \[A'B'\]  passing through P. If tangents to the circle at A and B meet at \[Q\,\,(h,\text{ }k),\] then locus of Q is called the polar of P with respect to circle and P is called the pole and if tangents to the circle at \[A'\] and \[B'\] meet at \[Q'\], then the straight line \[QQ'\] is polar with P as its pole.

Equation of polar of the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] w.r.t. \[({{x}_{1}},{{y}_{1}})\] is \[x{{x}_{1}}+y{{y}_{1}}+g(x+{{x}_{1}})+f(y+{{y}_{1}})+c=0\].

If the circle is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\], then its polar w.r.t. \[({{x}_{1}},{{y}_{1}})\]is \[x{{x}_{1}}+y{{y}_{1}}-{{a}^{2}}=0\].

The pole of the line \[lx+my+n=0\] with respect to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\].  Let pole be \[({{x}_{1}},\,{{y}_{1}}),\] then equation of polar with respect to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] is \[x{{x}_{1}}+y{{y}_{1}}-{{a}^{2}}=0\], which is same as \[lx+my+n=0\]

Then \[\frac{{{x}_{1}}}{l}=\frac{{{y}_{1}}}{m}=-\frac{{{a}^{2}}}{n}\], \[\therefore \] \[{{x}_{1}}=-\frac{{{a}^{2}}l}{n}\] and \[{{y}_{1}}=-\frac{{{a}^{2}}m}{n}\].

Hence, the required pole is \[\left( -\frac{{{a}^{2}}l}{n},-\frac{{{a}^{2}}m}{n} \right)\].

Properties of pole and polar

(i) If the polar of \[P\,({{x}_{1}},\,{{y}_{1}})\] w.r.t. a circle passes through \[Q\,({{x}_{2}},\,{{y}_{2}})\] then the polar of Q will pass through P and such points are said to be conjugate points.

(ii) If the pole of the line \[ax+by+c=0\] w.r.t. a circle lies on another line \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0;\] then the pole of the second line will lie on the first and such lines are said to be conjugate lines.