Question

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

• A. $y^{2}=ax \, $
• B. $y^{2}=2ax$
• C. $y^{2}=4ax$
• D. $x^{2}=4ay$

Answer 1

Answer: (B)

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