25. Point P(x, 4) lies on the line segment joining the points A-5, 8)

1.	25. Point P(x, 4) lies on the line segment joining the points A-5, 8) andB(4, -10). Find the ratio in which point P divides the line segment AB.Also find the value of x.na? RA - 100 AH als Fargus Peta
Answer» $\textbf{Given points are}$ $\text{A(-5,8) and B(4,-10)}$ $\text{since A,P and B are collinear, we have}$ $\text{Slope of AP=Slope of BP}$ $\frac{8-4}{-5-x}=\frac{4+10}{x-4}$ $\frac{4}{-5-x}=\frac{14}{x-4}$ $\frac{2}{-5-x}=\frac{7}{x-4}$ $2x-8=-35-7x$ $9x=-27$ $\implies\boxed{\bf\;x=-3}$ $\therefore\text{Point P is (-3,4)}$ $\text{Let the point P divides the line segment joining A and B in the ratio m:n}$ $\text{Then, the coordinates of P is given by}$ $(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})$ $(\frac{4m-5n}{m+n},\frac{-10m+8n}{m+n})$ $\text{But,}(\frac{4m-5n}{m+n},\frac{-10m+8n}{m+n})=(-3,4)$ $\implies\frac{4m-5n}{m+n}=-3\;\frac{-10m+8n}{m+n}=4$ $\implies\;7m=2n\;\text{and}\;-14m=-4n$ $\implies\frac{m}{n}=\fra{2}{7}$ $\implies\boxed{\bf\;m:n=2:7}$