First slide

Equation of line in Straight lines

Question

$Let A \equiv (3, - 4), B \equiv (1, 2) . Let P \equiv (2 k - 1, 2 k + 1) be a variable point such that PA + PB is the minimum. Then kis$

Moderate

A

$\frac{7}{9}$
B

0
C

$\frac{7}{8}$
D

9

Solution

$P A + P B i s m i n i m u m o n l y w h e n A, B, P a r e c o l l i n e a r I f t h r e e p o i n t s a r e c o l l i n e a r t h e n t h e s l o p e s o f t h e l i n e s j o i n i n g a n y t w o p o i n t s i s s a m e P A s l o p e i s \frac{2 k + 5}{2 k - 4} a n d A B s l o p e i s \frac{6}{- 2} E q u a t e t h e s l o p e s a n d t h e n s o l v e f o r k \frac{2 k + 5}{2 k - 4} = - 3 2 k + 5 = - 6 k + 12 8 k = 7 k = \frac{7}{8}$

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