The locus of the mid-points of all chords of | vedclass

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

Question

(D) Let the chord pass through the vertex $V(0, 0)$ and intersect the parabola $y^{2}=4ax$ at $P(at^{2}, 2at)$.Let $(h, k)$ be the mid-point of the ch

Vedclass · Accepted Answer

$x=-\frac{a}{2}$

Vedclass · Answer

(D) Let the chord pass through the vertex $V(0, 0)$ and intersect the parabola $y^{2}=4ax$ at $P(at^{2}, 2at)$.Let $(h, k)$ be the mid-point of the chord $VP$.Then,$h = \frac{at^{2}+0}{2} = \frac{at^{2}}{2}$ and $k = \frac{2at+0}{2} = at$.From $k = at$,we have $t = \frac{k}{a}$.Substituting $t$ in the expression for $h$: $h = \frac{a}{2} \left(\frac{k}{a}\right)^{2} = \frac{k^{2}}{2a}$.Thus,$k^{2} = 2ah$.The locus of the mid-point $(h, k)$ is $y^{2} = 2ax$.This can be rewritten as $(y-0)^{2} = 4\left(\frac{a}{2}\right)(x-0)$.For a parabola $Y^{2} = 4AX$,the directrix is $X = -A$.Here,$A = \frac{a}{2}$,so the directrix is $x = -\frac{a}{2}$.

The locus of the mid-points of all chords of the parabola $y^{2}=4ax$ passing through its vertex is another parabola with directrix:

Similar Questions

The locus of a point such that two tangents drawn from it to the parabola $y^2 = 4ax$ are such that the slope of one is double the other is:

What is the maximum number of normals that can be drawn from any interior point to a parabola?

If a chord of the parabola $y^2=4x$ passes through its focus and makes an angle $\theta$ with the $X$-axis,then its length is

If the focus of the parabola is $(3, -4)$ and the directrix is $y - 4 = 0,$ then the equation of the parabola is :-

If $x-2=t^2$ and $y=2t$ are the parametric equations of the parabola $y^2=a(x-b)$,then the value of $a+b$ equals

Vedclass Test Series

Exam Paper Generator

Online Exam Module