Determine the x-coordinate of the point A, where the ray intersect the upper surface of the slab air boundary

Q. determine the coordinates ($x_1,y_1$) of the point P, where the ray intersects the upper surface of the slab-air boundary.

Q. In the above situation if $y_0=a/2$, the coordinates of the point where the ray intersects the upper surface of the slabair boundary are

Q. A ray of light travelling in air is incident at angle of incidence, i$\approx {90}^o$ on a long rectangular slab of a transparent medium of thickness $y_0$. The medium has a variable refraction index of $\mu \left(x\right)=\sqrt{1+e^{{}^{2x}/{}_a}}\forall x\ge 0$, where a is a positive constant. If $y_0=\frac{a}{2}$, then $x$ coordinates of the point where the ray intersects the upper surface of the slab-air boundary is a $ln$ $\left(N\right)$. Find the value of $N$ . ​​​​​​​

Q. In the previous question, the angle made by light ray with +ve x-axis at the upper surface of a slab-air boundary, inside the medium is

Q. A ray of light travelling in air is incident at grazing angle (incident angle $={90}^o$) on a long rectangular slab of a transparent medium of thickness $t=1.0m$. The point of incidence is the origin A(0, 0). The medium has a variable index of refraction n(y) given by
$n\left(y\right)=[k{y}^{3/2}+1{]}^{1/2}$
where $k=1.0(m{)}^{-3/2}$
a. Obtain a relation between the slope of the trajectory of ray at point B(x, y) in the medium and the incident angle at the point.
b. Obtain an equation for the trajectory y(x) of the ray in the medium.
c. Determine the coordinates $(x_1,y_1)$ of point P, where the ray intersects the upper surface of the slab-air boundary.
d. Indicate the path of the ray subsequently