First slide

Equation of line in Straight lines

Question

Suppose that $A (5, 6)$ and $B (3, 4)$ be two points and $P$ be a point on $x -$ axis such that $|P A + P B|$ is minimum then the point $P$ is

Moderate

A

$(- 15, 5)$
B

$(- 10, 0)$
C

$(- 10, 10)$
D

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

Solution

$Suppose that the images of A (5, 6) and B (3, 4) in the x -axis are C and D respectively.$
$The point P is the point of intersection of the lines A D and x -axis, it is coincide with the$
$point of intersection of the lines B C and x -axis$
$The image of B (3, 4) in the x -axis is D (3, - 4)$
$The equation of the line joining D (3, - 4) and A (5, 6) is y - 6 = \frac{6 + 4}{5 - 3} (x - 5)$
This can be simplified as
$\begin{array}{l} y - 6 = \frac{6 + 4}{5 - 3} (x - 5) \\ y - 6 = 5 (x - 5) \\ y - 6 = 5 x - 25 \\ 5 x - y - 19 = 0 \end{array}$
Plug in $y = 0$ to get the point of intersection of the above line and x-axis.
It implies that
$\begin{matrix} 5 x = 19 \\ x = \frac{19}{5} \end{matrix}$
$Therefore, P = (\frac{19}{5}, 0)$

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