Question

If the point $C(-1,2)$ divides internally the line segment joining the points $A(2,5)$ and $B(x,y)$ in the ratio 3:4, find the value of $x^2+y^2$.

Answer 1

According to section formula, if P(x,y) divides the line through A(a,b) and B(c,d) in the ratio m:n

Then P is given by:

$P\left(x,y\right)=\left(\frac{mc+na}{m+n},\frac{md+nb}{m+n}\right)$

So,

$\begin{array}{c}C\left(-1,2\right)=\left(\frac{3\left(x\right)+4\left(2\right)}{3+4},\frac{3\left(y\right)+4\left(5\right)}{3+4}\right)\\ \left(-1,2\right)=\left(\frac{3x+8}{7},\frac{3y+20}{7}\right)\\ So,\\ \frac{3x+8}{7}=-1and\frac{3y+20}{7}=2\\ x=-5andy=-2\end{array}$

So,

$\begin{array}{c}{x}^2+y^2={\left(-5\right)}^2+{\left(-2\right)}^2\\ =25+4\\ =29\end{array}$