The abscissa of any two points on the parabola $y^2=4ax$ are in the ratio  $\mu :1$. If the locus of the point of intersection of tangents at these points is $y^2={\left({\mu }^{1/\lambda }+{\mu }^{-1/\lambda }\right)}^{2}ax$. Then the value of  $\lambda$=

Let P and Q be two points $t_1$ and $t_2$ respectively whose abscissas are in the ration m : 1.

$\therefore \frac{a{t}_1^2}{a{t}_2^2}=\frac{\mu }{1}$ or $t_1=t_2\sqrt{\left(\mu \right)}$

If (h, k) be the point of intersection of tangents at P and Q, then 

$h=a{t}_1{t}_2, k=a\left(t_1+t_2\right)$ Or $h=a\sqrt{\left(\mu \right)}{t}_2^{2}k=a\left(1+\sqrt{\mu }\right)t_2$

Eliminating $t_2$ we get the required locus as $y^2=ax{\left({\mu }^{1/4}+{\mu }^{-1/4}\right)}^2$