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 k is

Easy

A

7/9
B

0
C

7/8
D

None of these

Solution

We know that $PA + PB \geq AB$ (by triangle inequality). So, $PA + PB$ is the minimum if $PA + PB = AB$ , i.e., $A, P, B$ are collinear. Therefore,
$|\begin{matrix} 3 & - 4 & 1 \\ 1 & 2 & 1 \\ 2 k - 1 & 2 k + 1 & 1 \end{matrix}| = 0$
or $3 (2 - 2 k - 1) + 4 (1 - 2 k + 1) + 1 (2 k + 1 - 4 k + 2) = 0$
or $3 - 6 k + 8 - 8 k + 3 - 2 k = 0$
or $14 - 16 k = 0$
$∴ k = \frac{7}{8}$

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