Romeo and Juliet without Shakespeare!

To minimize the Distance Mathematician ( Romeo ) must Follow The Path of Light ( according to Fermat's Principle )

Now Take the image of $P(-3,4)$ . in the x-axis. Let The image is $S(-3,-4)$ .

Therefore $Q(0,1)\quad \& \quad R(\alpha ,0)\quad \& \quad { S }(-3,-4)$ . must be collinear.

Hence Slope :

${ \quad m }_{ SR }={ m }_{ SQ }\\ \Rightarrow \frac { -4-0 }{ -3-\alpha } =\frac { -4-1 }{ -3-0 } \\ \Rightarrow \quad \alpha =\frac { -3 }{ 5 } \\ \Rightarrow \quad \quad a=-3\quad \& \quad b=5$ .

NOTE

Principle which is Used in this Question is Fermat's Principle

Wow interesting! I knew there must be an easier method but I didn't know. I solved it by yet another method -- graphically. I just used a spreadsheet to plot the graph.

I also didn't know about the optical method. But I know now. I am giving the solution here.

The the angle of incident and angle of reflection to the normal at $R$ be $\theta_i$ and $\theta_r$ , then:

$\theta_i = \theta_r \quad \Rightarrow \tan {\theta_i} = \tan {\theta_r} \quad \Rightarrow \dfrac {x_Q - x_R} {y_Q - y_R} = - \dfrac {x_R - x_P} {y_R - y_P}$

$\Rightarrow \dfrac {0 - \alpha } {1 - 0} = - \dfrac {\alpha +3} {0 -4} \quad \Rightarrow 4\alpha = - \alpha -3 \quad \Rightarrow 5\alpha = -3 \quad \Rightarrow \alpha = - \dfrac {3}{5} = \dfrac {a}{b}$

Therefore, the required answer $=|a\times b| = |-3\times 5| = \boxed {15}$

While Fermat's principle would certainly be a much easier and straightfoward method to solve the question, It is also possible to solve it using calculus.( I found this question under the extrema section) The distance as a function of the point x is given by $\sqrt{(x+3)^2 + 4^2} +\sqrt{x^2 +1}$ We find the derivative of this function , set it to 0 and solve to find the critical points, 1 and - $\frac{3}{5}$ . By the second derivative test, minima occurs at the latter point and so the distance is minimum when $\boxed{x = \frac{-3}{5}}$

just think that the mathematician is a light wave and find the path such that he falls somewhere on the x-axis and is reflected towards his girl friends house,, In short fermats principle