I am solving some “nice” problems below:

Problem 1:

Show that the locus of a point that divides a chord of slope 2 of the parabola internally in the ratio is a parabola. Find the vertex of this parabola.

Solution 1:

Let and be the extremities of the chord with slope 2.

Hence,

Let be co-ordinates of the point which divides PQ in the ratio . Then,

and

and

and

Hence, the locus of is

, which is a parabola with vertex .

Problem 2:

If and are two focal chords of the parabola , then show that the chords and intersect on the directrix of the parabola.

Solution 2:

Let the co-ordinates of be for .

Since is a focal chord, —– call this Equation I.

Similarly, —- call this Equation II.

Equation of is —- call this Equation III.

and that of is —- call this Equation IV.

Using I and II, IV reduces to

that is, — call this Equation V.

Adding III and V we get:

, which in turn implies, , which in turn implies, that . Hence, III and V intersect on the directrix .

More later,

Nalin Pithwa.

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