First slide

Equation of line in Straight lines

Question

Consider the points $A (3, 4)$ and $B (7, 13) .$ If P is a point on the line $y = x$ such that $PA + PB$ is minimum, then the coordinates of P are

Moderate

A

$(12 / 7, 17 / 5)$
B

$(- 24 / 5, 17 / 5)$
C

$(31 / 7, 31 / 7)$
D

$(0, 0)$

Solution

Consider a point A' the image of A in $y = x .$ Therefore, the coordinates of $A'$ are $(4, 3)$ .
Notice that A and B lies on the same side with respect to $y = x .$ Then $PA = PB'$ is minimum, i,e., if $P, A'$ B are collinear. Now, the equation of A'B is
$y - 3 = \frac{13 - 3}{7 - 4} (x - 4)$
or $3 y - 10 x + 31 = 0$

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