Question

The sum of the reciprocal of focal distance of a focal chord of y2=4ax is

Answer 1

Answer:

Let y2 = 4ax be the equation of a parabola and (at2, 2at) a point P on it. Suppose the coordinates of the other extremity Q of the focal chord through P are (at12, 2at1).

Then, PS and SQ, where S is the focus (a, 0), have the same slopes

⇒ (2at-0)/(at2- a) = (2at1 - 0)/(at12 - a)

⇒ tt12 – t = t1t2 – t1

⇒ (tt1 + 1) (t1 – t) = 0.

Hence t1 = –1/t, i.e. the point Q is (a/t2, –2a/t).

The extremities of a focal chord of the parabola y2 = 4ax may be taken as the points t and –1/t.  

Step-by-step explanation: