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.