Let the locus of the mid-point of the chord t | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the parabola be $y^2=4x$. Any point on the parabola is $P(t^2, 2t)$.The chord passes through the origin $O(0,0)$ and $P(t^2, 2t)$.The mid-poin

Vedclass · Accepted Answer

$2y^2=3x$

Vedclass · Answer

(B) Let the parabola be $y^2=4x$. Any point on the parabola is $P(t^2, 2t)$.The chord passes through the origin $O(0,0)$ and $P(t^2, 2t)$.The mid-point $M(h, k)$ of the chord $OP$ is given by $h = \frac{t^2+0}{2} = \frac{t^2}{2}$ and $k = \frac{2t+0}{2} = t$.Substituting $t=k$ into $h = \frac{t^2}{2}$,we get $h = \frac{k^2}{2}$,so $k^2=2h$.The locus $S$ is $y^2=2x$.Let $P(x_0, y_0)$ be a point on $S$,so $y_0^2=2x_0$. Since $P$ is on $S$,we can write $P$ as $(2t^2, 2t)$ because $(2t)^2 = 2(2t^2)$.Let $R(x, y)$ be the point that divides $OP$ internally in the ratio $3:1$. Using the section formula:$x = \frac{3(2t^2) + 1(0)}{3+1} = \frac{6t^2}{4} = \frac{3t^2}{2}$$y = \frac{3(2t) + 1(0)}{3+1} = \frac{6t}{4} = \frac{3t}{2}$From $y = \frac{3t}{2}$,we get $t = \frac{2y}{3}$.Substituting $t$ into $x = \frac{3t^2}{2}$:$x = \frac{3}{2} \left(\frac{2y}{3}\right)^2 = \frac{3}{2} \cdot \frac{4y^2}{9} = \frac{2y^2}{3}$Thus,$3x = 2y^2$,or $2y^2=3x$.

Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^{2}=4x$ be the curve $S$. Let $P$ be any point on $S$. Then the locus of the point,which internally divides $OP$ in the ratio $3:1$,is:

Similar Questions

If the line $2bx + 3cy + 4d = 0$ passes through the points of intersection of $y^2 = 4ax$ and $x^2 = 4ay$,then

If the normals drawn at the points $P\left(\frac{3}{4}, \frac{3}{2}\right)$ and $Q(3,3)$ on the parabola $y^2=3x$ intersect again on the parabola at $R$,then $R=$

For the parabola $y = \frac{h^3}{3} x^2 + \frac{h^2}{2} x - h + \frac{3}{4 h^3}$,if the equation of the directrix is $y = k$,then find the ratio $k : h$.

Find the equation of the latus rectum of the parabola $x^2 = -12y$.

If the tangents and normals at the extremities of a focal chord of a parabola $y^2 = 4ax$ intersect at $(x_1, y_1)$ and $(x_2, y_2)$ respectively,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module