Pole of a Line

The point whose polar is the given line. Pole and polar are reciprocal: if Q lies on the polar of P, then P lies on the polar of Q (La Hire's theorem).

The pole of a line $L$ with respect to a circle is the unique point $P$ such that $L$ is the polar of $P$.

For $x^2 + y^2 = a^2$: The polar of $(x_1, y_1)$ is $xx_1 + yy_1 = a^2$.

Given line $lx + my + n = 0$, rewrite as $lx + my = -n$. Compare with $xx_1 + yy_1 = a^2$:

$$\frac{x_1}{l} = \frac{y_1}{m} = \frac{a^2}{-n}$$ $$x_1 = -\frac{al}{n} \cdot a = -\frac{a^2 l}{n}, \quad y_1 = -\frac{a^2 m}{n}$$

Therefore the pole of $lx + my + n = 0$ with respect to $x^2 + y^2 = a^2$ is:

$$\left(-\frac{a^2 l}{n},\; -\frac{a^2 m}{n}\right)$$

La Hire's theorem (reciprocity): $P$ is the pole of $L$ if and only if $L$ is the polar of $P$. Consequently:

• If $P$ lies on $L$, then the pole of $L$ lies on the polar of $P$
• Two conjugate lines (each passing through the pole of the other) form a harmonic range with the circle

Application: In JEE problems, pole-polar relationships often appear in locus problems involving tangent lines, or when finding the equation of the chord of contact given the intersection of tangents.