Equation of the parabola is `y^2 = 4ax`..........................................(1)

Let `AB` be a chord of the parabola whose middle point is `P (x_1,y_1)`.

Equation of chord `AB` is `y - y_1 = m (x - x_1)`.....................................(2)

where `m =` slope of `AB`

Let `A = (x_2, y_2)` and `B = (x_3, y_3)`.

Since `A` and `B` lie on parabola `(1)`

`:. y_2^2=4ax_2` and `y_3^2=4ax_3`

`:. y_2^2-y_3^2=4a(x_2-x_3)` or `(y_2-y_3)/(x_2-x_3)=(4a)/(y_2+y_3)` .....................(3)

But `P(x_1,y_1)` is the middle point of `AB` `y_2 + y_3 = 2y_1`

`:. ` From (3), `(y_2-y_3)/(x_2-x_3) =(4a)/(2y_1)=(2a)/(y_1)`

`:. ` Slope of `AB` i.e., `m = (2a)/(y_1)`

From (2), equation ofchord `AB` is `y - y_1=(2a)/(y_1) (x-x_1)`

or `yy_1- y_1^2 = 2ax - 2ax_1` or `yy_1 - 2ax = y_1^2 - 2ax_1`

or `yy_1-2a(x-x_1)=y_1^2-4ax_1` [Subtracting `2ax_1` from both sides].............................(5)

(5) is the required equation. In usual notations, equation (5) can be written as `T = S_1`.

The same result holds true for cicle, ellipse and hyperbola also.

Two tangents `PA` and `PB` are drawn to parabola, then line joining `AB` is called the chord of contact to the
parabola with respect to point `P`.

Let the parabola be `y^2 = 4ax` ....................(1)

Let `P( alpha , beta)` be a point outside the parabola.

Let `PA` and `PB` be the two tangents from `P(alpha, beta)` to parabola `(1)`.

Let `A = (x_1, y_1)` and `B = (x_2, y_2)`

Equation of the tangent `PA` is `yy_1 = 2a (x + x_1)` ........................(2)

Equation of the tangent `PB` is `yy_2 = 2a (x + x_1)` ......................(3)

Since lines `(2)` and `(3)` pass through `P(alpha, beta)`, therefore

`beta y_1 = 2a(a + x_1 )` .........................(4)

and `beta y_2 = 2a (a + x_2)` ..................(5)

Now we consider the equation `y beta = 2a (x + alpha)` .....................(6)

From `(4)` and `(5)`, it follows that line `(6)` passes through `A (x_1, y_1)` and `B (x_2, y_2)`.

Hence `(6)` is the equation of line `AB` which is the chord of contact of point `P(alpha , beta)` with respect to parabola `(1)` i.e, chord of contact is `y beta = 2a (x +alpha)`

The same result holds true for circle, ellipse and hyperbola also.