Question

In the point $C(1,1)$ divides the line segment joining $A(-2,7)$ and B in the ratio $3:2$, find the coordinates of B.

Answer 1

We know that the section formula states that if a point $P(x,y)$ lies on line segment $AB$ joining the points $A(x_1,y_1)$ and $B(x_2,y_2)$and satisfies $AP:PB=m:n$, then we say that $P$ divides internally $AB$ in the ratio $m:n$. The coordinates of the point of division has the coordinates

$P=(\frac{{mx}_2{+nx}_1}{m+n},\frac{{my}_2{+ny}_1}{m+n})$

Let $C(1,1)$ divides the line segment $AB$ joining the points $A(-2,7)$ and $B(x_2,y_2)$ in the ratio $3:2$, then using section formula we get,

$C=(\frac{{mx}_2{+nx}_1}{m+n},\frac{{my}_2{+ny}_1}{m+n})\Rightarrow (1,1)=(\frac{{3x}_2+(2\times -2)}{3+2},\frac{{3y}_2+(2\times 7)}{3+2})\Rightarrow (1,1)=(\frac{{3x}_2-4}{5},\frac{{3y}_2+14}{5})\Rightarrow 1=\frac{{3x}_2-4}{5},1=\frac{{3y}_2+14}{5}$
$\Rightarrow 5={3x}_2-4,5={3y}_2+14\Rightarrow {3x}_2=5+4,{3y}_2=5-14\Rightarrow {3x}_2=9,{3y}_2=-9\Rightarrow x_2=\frac{9}{3},y_2=-\frac{9}{3}\Rightarrow x_2=3,y_2=-3$

Hence, the point $B(x_2,y_2)$ is $B(3,-3)$.