A particle p moves from the point $$A(0$$,$$4)$$ to the point (10$$,−$$4). The particle P can travel the $$upper-half$$ plane $${(x$$,$$y)$$|y≥$$0}$$ at the speed of $$1$$ $$m/s$$ and the $$lower-halfplane{(x$$,$$y)$$|y≤$$0}$$ at the speed of $$2$$ $$m/s$$. The coordinates of a point on the $$x-axis$$, if the sum of the squares of the travel times of the $$upper-$$ and $$lower-half$$ planes is minimum, are

Question

A particle p moves from the point $$A(0$$,$$4)$$ to the point (10$$,−$$4). The particle P can travel the $$upper-half$$ plane $${(x$$,$$y)$$|y≥$$0}$$ at the speed of $$1$$ $$m/s$$ and the $$lower-halfplane{(x$$,$$y)$$|y≤$$0}$$ at the speed of $$2$$ $$m/s$$. The coordinates of a point on the $$x-axis$$, if the sum of the squares of the travel times of the $$upper-$$ and $$lower-half$$ planes is minimum, are

Infinity Learn · Accepted Answer

Let the point on the $$x-axis$$ be (c$$,$$0). Sum of the squares of travel times $$isT=(c2+161)2+((10$$−$$c)2+162)2=c2+16+116+c2$$−$$20c4=54c2$$−$$5c+45=54(c2$$−$$4c+36)Therefore$$, T is minimum at $$c=2$$

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