Let `P` be any point inside or outside a parabola. Suppose in a straight line drawn through `P` intersect the parabola at `Q` and `R`. Then the locus of point of intersection of the tangents to the parabola at `Q` and `R` is called the polar of given point `P` with respect to the parabola and point `P` is called the pole of the polar.

Equation of the Polar of the point `P(x_1, y_1)` w.r.t. the parabola `y^2 = 4ax` is,

`yy_1=2a(x-x_1)`

The pole of the line `lx + my + n = 0` w.r.t. the parabola `y^2= 4ax` is `(n/l,-(2am)/l)`.


Properties of polar :

(i) The polar of the focus of the parabola is the directrix .

(ii) When the point `(x_1, y_1)` lies without the parabola the equation to its polar is the same as the equation to the chord of contact of tangents drawn from `(x_1, y_1)` when `(x _1,y_1)` is on the parabola the polar is the same as the tangent at the point.

(iii) Two straight lines are said to be conjugated to each other w.r.t. a parabola when the pole of one lies on the other. Similarly two points P and `Q` are said to be conjugate points if polar of `P` passes through `Q` and vice versa.

(iv) Polar of a given point `P` w.r.t. any Conic is the locus of the harmonic conjugate of `P` w.r.t. the two points is which any line through P cuts the conic.