Some problems on conics (parabola, ellipse, hyperbola) : IITJEE Mains — Basics 2

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 y^{2}=4x internally in the ratio 1:2 is a parabola. Find the vertex of this parabola.

Solution 1:

Let P(t_{1}^{2}, 2t_{1}) and Q(t_{2}^{2}, 2t_{2}) be the extremities of the chord with slope 2.

Hence, \frac{2t_{1}-2t_{2}}{t_{1}^{2}-t_{2}^{2}}=2 \Longrightarrow t_{1}+t_{2}=1

Let (t_{1},t_{2}) be co-ordinates of the point which divides PQ in the ratio 1:2. Then,

h=\frac{2t_{1}^{2}+t_{2}^{2}}{3} and k=\frac{4t_{1}+2t_{2}}{3}

\Longrightarrow 3h = 2t_{1}^{2}+(1-t_{1})^{2} and 3k = 4t_{1}+2(1-t_{1})

\Longrightarrow 3h = 3t_{1}^{2} - 2t_{1} + 1 and 3k = 2t_{1}+2

\Longrightarrow 3h = 3(\frac{3k-2}{2})^{2} -2(\frac{4k-2}{2})+1

\Longrightarrow 12h = 3(9k^{2}-12k+4)-12k+8+4

4h = 9k^{2}-16k+8

Hence, the locus of (h,k) is 9y^{2}-16y-4x+8=0

\Longrightarrow (3y-\frac{8}{3})^{2}=4x-8+\frac{64}{9}

\Longrightarrow (y-\frac{8}{9})^{2}=\frac{4}{9}(x-\frac{2}{9}), which is a parabola with vertex (\frac{2}{9}, \frac{8}{9}).

Problem 2:

If P_{1}P_{2} and P_{3}P_{4} are two focal chords of the parabola y^{2}=4ax,  then show that the chords P_{1}P_{3} and P_{2}P_{4} intersect on the directrix of the parabola.

Solution 2:

Let the co-ordinates of P_{i} be (at_{i}^{2},2at_{i}) for i=1,2,3,4.

Since P_{1}P_{2} is a focal chord, t_{1}t_{2}=-1 —– call this Equation I.

Similarly, t_{3}t_{4}=-1 —- call this Equation II.

Equation of P_{1}P_{3} is y(t_{1}+t_{3})=2(x+at_{1}t_{3}) —- call this Equation III.

and that of P_{2}P_{4} is y(t_{2}+t_{4})=2(x+\alpha t_{2}t_{4}) —- call this Equation IV.

Using I and II, IV reduces to y(-\frac{1}{t_{1}}-\frac{1}{t_{2}})=2(x+\frac{a}{t_{1}}t_{3})

that is, -y(t_{1}+t_{3})=2(xt_{1}t_{3}+a) — call this Equation V.

Adding III and V we get:

0=2[x+at_{1}t_{3}+xt_{1}t_{3}+a], which in turn implies, (x+a)(1+t_{1}t_{3})=0, which in turn implies, that x=-a. Hence, III and V intersect on the directrix x+a=0.

More later,

Nalin Pithwa.

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