Properties of the parabola:
(i) The tangent at any point P on the parabola bisects angle between the focal chord through P and the perpendicular from P on directrix.
In the given figure ∠MPT = ∠TPS :
Likewise the normal at any point on a parabola bisects angle between the focal chord and the line parallel to axis through that point.
(ii) The portion of a tangent to the parabola cut off between directrix and the curve subtends a right angle at focus.
In the given figure SP is perpendicular to SK that is ∠KSP = 900 .
(iii) Tangents at extremities of any focal chord intersect at the right angles on directrix.
(iv) Any tangent to parabola and perpendicular on it from focus meets on the tangent at vertex.
Pole and Polar
To find out the equation of polar of the point (x1, y1) with respect to parabola y2 = 4ax.
Let Q and R be points in which any chord drawn through point P, whose coordinates are (x1, y1), meets the parabola.
Suppose that the tangents at Q and R meet in point whose coordiantes are (h, k).
We require locus of (h, k).
As QR is the chord of contact of tangents from (h, k) the equation of it is ky = 2a (x + h)
As this straight line passes through point (x1, y1) we have
ky1 = 2a (x 1+ h) ....(i)
As the relation (i) is true, it follows that point (h, k) always lies on straight line
yy1 = 2a(x + x1) ....(ii)
Thus (ii) is the equation to polar of (x1, y1).
Example: Prove that locus of poles of focal chord of parabola y2 = 4ax is the directrix.
Solution: Let (h, k) be pole. Then the equation of chord is ky = 2a(x + h).
As it is a focal chord, it passes through the focus (a, 0).
∴ 2a(a + h) = 0 or a + h = 0
Thus locus of pole (h, k) is
x + a = 0, which is directrix.
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