In a flat race, A beats B by 15 m and C by 29 m. When B and C run over the same course together, B wins by 15 m. The length of the course is

Let the race course of length be "L" . So lets assume time taken by A to finish it be "t" seconds . 
                              $$t=\dfrac{\ L}{V_a}$$

$$V_b(t)=V_b[\frac{\ L}{V_a}]=L-15$$ ... equation 1
$$V_c(t)=V_c[\frac{\ L}{V_a}]=L-29$$ .....equation 2

In race between B & C for length of L, B wins by 15 m : 
                                     $$V_b(t_2)=L$$...........equation 3
                                    $$V_c(t_2)=L-15$$.........equation 4 

Dividing equation 1 by equation 2 : $$\dfrac{\ V_b}{V_c}=\dfrac{\ L-15}{L-29}$$ 

Dividing equation 3 by equation 4 : $$\dfrac{\ V_b}{V_c}=\dfrac{\ L}{L-15}$$ 

         So, $$\dfrac{\ L-15}{L-29}$$ = $$\dfrac{\ L}{L-15}$$

         it implies ,  $$(L-15)^2=L^2-29L$$
                           L = 225.

Hence, option C.