Question

Prove that for all values of m, except zero the straight line $y=mx+\frac{a}{m}$ touches the parabola $y^2=4ax$.

Answer 1

The equation of line and parabola are
$y=mx+\frac{a}{m}$....(i)
and $y^2=4ax$....(ii)
On solving Eqs. (i) and (ii), we get
$y^2-\frac{4a}{m}y+\frac{4a^2}{m^2}=0$
$\Rightarrow {(y-\frac{2a}{m})}^2=0$
$\Rightarrow y=\frac{2a}{m},\frac{2a}{m}$
$\because$ Both values of y are same. Therefore, line touch the parabola at all point except $m=0$.