The locus of the middle points of the chords of the parabola y2=4ax, which passes through the origin is

Question

The locus of the middle points of the chords of the parabola y2=4ax, which passes through the origin is (A) y2=ax  (B) y2=2ax (C) y2=4ax (D) x2=4ay

Tardigrade Admin · Accepted Answer

Let mid point be (.x1,y1.) ∴ Equation of chord is T=S1 (y y)1-2a(x + x1)=y12-4ax1 Since, it passes through origin ∴ -2ax1=y12-4ax1 ⇒ y12=2ax1 ∴ Locus is y2=2ax

Q. The locus of the middle points of the chords of the parabola $y^{2} = 4 a x,$ which passes through the origin is

$y^{2} = a x$

$y^{2} = 2 a x$

$y^{2} = 4 a x$

$x^{2} = 4 a y$

Let mid point be $(x_{1}, y_{1})$
$∴$ Equation of chord is
$T = S_{1}$
$(yy)_{1} - 2 a (x + x_{1}) = y_{1}^{2} - 4 a x_{1}$
Since, it passes through origin
$∴ - 2 a x_{1} = y_{1}^{2} - 4 a x_{1}$
$\Rightarrow y_{1}^{2} = 2 a x_{1}$
$∴$ Locus is $y^{2} = 2 a x$

Q. The locus of the middle points of the chords of the parabola y2=4ax, which passes through the origin is

y2=ax

y2=2ax

y2=4ax

x2=4ay