Question

25. Point P(x, 4) lies on the line segment joining the points A-5, 8) and
B(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 1

$\textbf{Given points are}$

$\text{A(-5,8) and B(4,-10)}$

$\text{since A,P and B are collinear, we have}$

$\text{Slope of AP=Slope of BP}$

$\frac{8-4}{-5-x}=\frac{4+10}{x-4}$

$\frac{4}{-5-x}=\frac{14}{x-4}$

$\frac{2}{-5-x}=\frac{7}{x-4}$

$2x-8=-35-7x$

$9x=-27$

$\implies\boxed{\bf\;x=-3}$

$\therefore\text{Point P is (-3,4)}$

$\text{Let the point P divides the line segment joining A and B in the ratio m:n}$

$\text{Then, the coordinates of P is given by}$

$(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})$

$(\frac{4m-5n}{m+n},\frac{-10m+8n}{m+n})$

$\text{But,}(\frac{4m-5n}{m+n},\frac{-10m+8n}{m+n})=(-3,4)$

$\implies\frac{4m-5n}{m+n}=-3\;\frac{-10m+8n}{m+n}=4$

$\implies\;7m=2n\;\text{and}\;-14m=-4n$

$\implies\frac{m}{n}=\fra{2}{7}$

$\implies\boxed{\bf\;m:n=2:7}$

Answer 2

Answer:

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