# For the parabola, y2 = 4px find the extremities of a double ordinate of length 8p. Prove that the lines from the vertex to its extremities are at right angles.

Let AB be the double ordinate of length 8p for the parabola y2 = 4px.

Comparing with standard form we get y2 = 4ax we get,

axis is y = 0

vertex is O(0, 0).

We know that double ordinate is perpendicular to the axis.

Let us assume that the point at which the double ordinate meets the axis is (x1, 0).

Then the equation of the double ordinate is y = x1. It meets the parabola at the points (x1, 4p) and (x1, - 4p) as its length is 8p.

Let us find the value of x1 by substituting in the parabola.

(4p)2 = 4p(x1)

x1 = 4p.

The extremities of the double ordinate are A(4p, 4p) and B(4p, - 4p).

Let us find assume the slopes of OA and OB be m1 and m2. Let us find their values.

m1 = 1

m2 = - 1

m1.m2 = 1. - 1

m1.m2 = - 1

We have got product of slopes - 1. So, the lines OA and OB are perpendicular to each other.

So, the extremities of double ordinate make right angle with vertex.

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