Question

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

Answer 1

We know that by section formula, the co-ordinates of the points which divide internally the line segment joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$ is

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Since, C divides A & B in the ratio $3:4$ internally

By section formula

$\frac{3x+4\left(2\right)}{3+4}=-1$, $\frac{3y+4\left(5\right)}{3+4}=2$

$\Rightarrow 3x+8=-7$, $3y+20=14$

$x=\frac{-15}{3}$, $3y=-6$

$\Rightarrow x=-5$, $y=-2$

$x^2+y^2=(-5{)}^2+(-2{)}^2$

$=25+4$

$\therefore x^2+y^2=29$.