Question

Inspired by the 'Golden quadrilateral project', UP Government recently accomplished a diamond triangular project. Under this project, the State Government laid down 6 lane roads connecting three cities Ayodhya, Banaras and Chitrakoot, which are equally separated from each other i. e, in terms of geometry they form an equilateral triangle. Angad and Bajrang start simultaneously from Ayodhya and Banaras respectively, towards Chitrakoot. When Angad cover 100 km, Bajrang covers such a distance that the distance between Angad and Bajrang makes ${90}^{\circ }$ angle with the road joining Banaras and Chitrakoot. When Bajrang reaches Chitrakoot, Angad is still 150 km away from Chitrakoot. What is the distance between Ayodhya and Banaras?

• A. 250 km
• B. 450 km
• C. 300 km
• D. none of these

Answer 1

The correct option is C 300 km


$\frac{100+\frac{x}{2}}{100}=\frac{100+x}{(100+x-150)}=\frac{SpeedofBajrang}{SpeedofAngad}\frac{200+x}{200}=\frac{(100+x)}{(x-50)}\Rightarrow (200+x)(x-50)=200(100+x)\Rightarrow x^2+150x-10000=20000+200x\Rightarrow x^2-50x-30000=0\Rightarrow (x-200)(x+150)=0\Rightarrow x=200km$
Therefore distance between Ayodhya and Banaras is 300 km since AB =BC = AC.
(Withe the help of trigonometry we can find the value of PC in terms x i.e., cos ${60}^{\circ }=\frac{PC}{QC}=\frac{1}{2}.HencePC=\frac{x}{2}$