91Ó°ÊÓ

The given parameterize position of points are $A=\left(x_1,y_1\right),B=\left(x_2,-y_2\right)$, and${∫}_A^{B}nds=n_1{d}_1+n_2{d}_2$.

"The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, for light of a given colour and for a given set of media," according to Snell's law. The formula for Snell's law is: $\frac{\sin i}{\sin r}=\mathrm{constant}$.

Let’s start with the principle of Fermat’s.

$$\begin{gathered} t=∫dt \\ =\frac{n_1}{c}{∫}_A^{P}ds+\frac{n_2}{c}{∫}_P^{B}ds......\left(1\right) \end{gathered}$$

Evaluate the equation (1) to calculate the distance between points $A=\left(x_1,y_1\right)$,

$B=\left(x_2,-y_2\right)\mathrm{and}P=\left(x,0\right)$.

$$\begin{gathered} {∫}_A^{P}ds=\sqrt{{\left(x-x_1\right)}^2+y_1^2}......\left(2\right) \\ {∫}_P^{B}ds=\sqrt{{\left(x_2-x\right)}^2+y_2^2}......\left(3\right) \end{gathered}$$

Substitute equations(2) and (3) into equation(1) and setting derivative to zero.

$$\begin{gathered} t=\frac{n_1}{c}\left(\sqrt{{\left(x-x_1\right)}^2+y_1^2}+\sqrt{{\left(x_2-x\right)}^2+y_2^2}\right).....\left(4\right) \\ \frac{dt}{dx}=0 \\ {n}_1\frac{x-x_1}{\sqrt{{\left(x-x_1\right)}^2+y_1^2}}=n_2\frac{x_2-x}{\sqrt{{\left(x_2-x\right)}^2+y_2^2}}......\left(5\right) \end{gathered}$$

Derive the following from the figure below.

Substitute equations(5) and (6) into equation(4).

$n_1\sin {\mathrm{θ}}_1=n_2\sin {\mathrm{θ}}_2$

Hence, proved.