First slide

Equation of line in Straight lines

Question

Suppose that $A (5, 1)$ and $B (7, 5)$ be two points and $P$ be a point on $y = x$ such that $|P A - P B|$ is minimum then the point is $P$

Moderate

A

$(- 15, 5)$
B

$(9, 9)$
C

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

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

Solution

$Since the condition is | P A - P B | minimum, the point P lies on the line A B$
$The equation of A B is$
$\begin{matrix} y - 5 = \frac{5 - 1}{7 - 5} (x - 7) \\ y - 5 = 2 x - 14 \\ 2 x - y - 9 = 0 \end{matrix}$
$Hence the point P is the point of intersection of the line 2 x - y - 9 = 0 and y = x$
$Plug in y = x in the equation 2 x - y - 9 = 0$
$\begin{matrix} 2 x - x - 9 = 0 \\ x = 9 \end{matrix}$
$Therefore, the point P is P (9, 9)$

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