First slide

Equation of line in Straight lines

Question

Suppose that $A (1, 2)$ and $B (3, 4)$ be two points and $P$ be a point on $y = x$ such that $|P A + P B|$ is minimum then the point $P$

Moderate

A

$(- 15, 5)$
B

$(- 10, 0)$
C

$(\frac{5}{2}, \frac{5}{2})$
D

$(- \frac{19}{5}, 0)$

Solution

$Suppose that the images of A (1, 2) and B (3, 4) in the line x = y are C and D respectively.$
$The point P is the point of intersection of the lines A D and x = y, it is coincide with the$
$point of intersection of the lines B C and x = y$
$The image of B (3, 4) in the line y = x is D (4, 3)$
$The equation of the line joining D (4, 3) an A (1, 2) is y - 2 = \frac{2 - 3}{1 - 4} (x - 1)$
This can be simplified as
$\begin{matrix} y - 2 = \frac{1}{3} (x - 1) \\ 3 y - 6 = x - 1 \\ x - 3 y + 5 = 0 \end{matrix}$
Plug in $x = y$ to get the point of intersection of the above line and the line $x = y$ .
It implies that
$\begin{matrix} x - 3 x + 5 = 0 \\ 2 x = 5 \\ x = \frac{5}{2} \end{matrix}$
$Therefore, P = (\frac{5}{2}, \frac{5}{2})$

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