Question

What is the distance between the points which divide the line segment joining $(4,3)$ and $(5,7)$ internally and externally in the ratio $2:3$?

• A. $\frac{12\sqrt{17}}{5}$
• B. $\frac{13\sqrt{17}}{5}$
• C. $\frac{\sqrt{17}}{5}$
• D. $\frac{6\sqrt{17}}{5}$

Answer 1

The correct option is A $\frac{12\sqrt{17}}{5}$

Given $(4,3)$ and $(5,7)$

Internally and externally in ratio

(1) Internal division:

$x=\frac{2\left(5\right)+3\left(4\right)}{5},y=\frac{2\left(7\right)+3\left(3\right)}{2+3}\to (x,y)=(\frac{22}{5},\frac{23}{5})$

(2) External division:

$x=\frac{2\left(5\right)-3\left(4\right)}{2-3},y=\frac{2\left(7\right)-3\left(3\right)}{2-3}\to (x,y)=(2,-5)$

$Distance=\sqrt{{(\frac{22}{5}-(2\left)\right)}^2+{(\frac{23}{5}-(5\left)\right)}^2}=\sqrt{\frac{144}{25}+\frac{2304}{25}}=\frac{12\sqrt{17}}{5}$.