Q. 55.0( 4 Votes )

# For the parabola, y^{2} = 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.

Answer :

Let AB be the double ordinate of length 8p for the parabola y^{2} = 4px.

Comparing with standard form we get y^{2} = 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 (x_{1}, 0).

Then the equation of the double ordinate is y = x_{1}. It meets the parabola at the points (x_{1}, 4p) and (x_{1}, - 4p) as its length is 8p.

Let us find the value of x_{1} by substituting in the parabola.

⇒ (4p)^{2} = 4p(x_{1})

⇒ x_{1} = 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 m_{1} and m_{2}. Let us find their values.

⇒

⇒

⇒ m_{1} = 1

⇒

⇒

⇒ m_{2} = - 1

⇒ m_{1}.m_{2} = 1. - 1

⇒ m_{1}.m_{2} = - 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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