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Local Minima

Consider poin...

Question

Consider points $A (3, 4)$ and $B (7, 13)$ . If P be a point on the line y =x such that PA +PB is minimum,then coordinates of P are

(\frac{12}{7}, \frac{12}{7})

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(\frac{13}{7}, \frac{13}{7})

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(\frac{31}{7}, \frac{31}{7})

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(0, 0)

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Solution

The correct option is A $(\frac{12}{7}, \frac{12}{7})$
$\begin{matrix} A c c o r d i n g t o q u e s t i o n . . . . . . . . . . . c o n s i d e r t h e p o i n t, A (3, 4), B (7, 13) a n d, I n y = x, P i s p o i n t & c o r d i n a t e o f p (a, a) s u c h t h a t, P A + P B \to M i n . s o, w e f i n d : P A = P A^{^{'}} N o w, w e c a n w r i t e a s : P A + P B = P A^{^{'}} + P B \to min (i f w e h a v e 3 l i n e a r p o i n t s a r e i n o n e l i n e, t h e n w e g e t m i n . d i s t a n c e,) N o w, F i n d e q u^{n} - A^{^{'}} B A^{^{'}} (4, 3), B (7, 13) l i n e o f A^{^{'}} B \Rightarrow y - 3 = \frac{13 - 3}{7 - 4} (x - 4) y - 3 = \frac{10}{7 - 4} (x - 4) 3 y - 9 = 10 x - 40 3 y - 10 x + 31 = 0 | I n t h i s e q u . . p (a, a) i s c o - l i n e a r t h e n, 3 a - 10 a + 31 = 0 a = \frac{31}{7} s o t h a t p o i n t P (\frac{31}{7}, \frac{31}{7}) & t h e c o r r e c t o p t i o n i s C . \end{matrix}$

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