**Properties of Tangents – Parabola**

**(i) Point of Intersection of Tangent at any Two Points on the Parabola.**

Let the equation of the parabola be y² = 4ax.

The two points on the parabola are P ≡ (at₁², 2at₁) and Q ≡ (at₂², 2at₂).

The equation of the tangents at P and Q are, respectively,

t₁y = x + at₁²

t₂y = x + at₂²

solving these equations, we get

x = at₁t₂

y = a(t₁ + t₂)

thus, the coordinates of the point of intersection of tangents at (at₁², 2at₁) and (at₂², 2at₂) are

(at₁ t₂, a(t₁ + t₂) ).

**(ii) Locus of Foot of Perpendicular from Focus upon any Tangent is Tangent at Vertex:**

The equation of the tangent to the parabola y² = 4ax at point P(t) is ty = x+ at².

It meets the y- axis at Q(0, at) Now m_{SQ} = (at – 0)/ (0 – a) = -t

The slope of tangent PQ is 1/t