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Q.
The locus of the middle points of the normàl chords of the parabola, $y^{2}=4ax$ is
NTA AbhyasNTA Abhyas 2022
Solution:
Here, equation of the normal chord at any point $a t^2, 2 a t$ ) of the parabola is
$y+t x=2 a t+a t^3 \ldots$ (i)
Equation of the chord with mid-point $x_1, y_1$ is $T=S_1 $
$y y_1-2 a x+x_1=y_1^2-4 a x_1 $
$y y_1-2 a x=y_1^2-2 a x_1 \ldots . \text { (ii) }$
since, Eqs. (i) and (ii) are identical
$\frac{1}{y_1}=\frac{t}{-2 a}=\frac{2 a t+a t^3}{y_1^2-2 a x_1}$
$t=\frac{-2 a}{y_1}$ and $\frac{y_1^2-2 a x_1}{-2 a}=\frac{2 a t+a t^3}{t}$
$=2 a+a \frac{-2 a^2}{y_1} $
$\text { or } \frac{-y_1^2}{2 a}+x_1=2 a+\frac{4 a^3}{y_1^2} $
$\Rightarrow x_1-2 a=\frac{y_1^2}{2 a}+\frac{4 a^3}{y_1^2}$
Hence, the locus of the middle point $x_1, y_1$ is
$x -2 a =\frac{ y ^2}{2 a }+\frac{4 a ^3}{ y ^2}$