# Parabola (S .N. Dey ) | Ex-4 | Part-3

In the previous article , we have solved 6 Long Answer Type Questions (7-12) of Parabola Chapter. In this article, we have solved 6 more Long answer type questions of Parabola Chapter (Ex-4) of S.N.Dey mathematics, Class 11.

##### is a double ordinate of the parabola find the equation to the locus of its point of trisection.

Solution.

Since is a double ordinate of the parabola let

Let the chord is trisected at the points and , where

So, by we get,

Hence, the equation of the required locus of the point of trisection is

Show that the circle described on a focal chord of a parabola as diameter touches its directrix.

Solution.

Let the equation of the parabola be

We know the extremities of the focal chord of the parabola be and .

the equation of the circle having diameter as the focal chord with the aforesaid extremities is

Now, the equation of the directrix of the parabola is

From and we get,

So, the point of intersection of the straight line and the circle is

Since has only one value, so we can say that the circle touches its directrix.

Prove that the sum of the reciprocals of the segments of any focal chord of a parabola is constant.

Solution.

Let the equation of the parabola be and be the focal chord of the parabola.

The length of latus rectum of a parabola is unit. The distance of a point on the parabola from its axis is unit. Find the distance of from the focus of the parabola.

Solution.

Let the equation of the parabola :

The length of the latus rectum ,

Any point on the parabola can be written as

By question,

The distance between the focus and

Prove that the length of any chord of the parabola passing through the vertex and making an angle with the positive direction of the axis is

Solution.

Let be the chord passing through the vertex where

By question, the slope of

the length of the chord

Here,

If and the line is passing through the points of intersection of parabolas and then prove that .

Solution.

The points of intersection of the parabolas and are and

Now, if the straight line passes through  the point , then   which is impossible as

So, the straight line passes through  the point .