Question

The midpoint p of the line segment joining the points A(-10,4) and B(-2,0) lies on the line segment joining the points C(-9,-4) and D(-4,y). find the ratio in which p divides CD.Also find the value of y.​

Answer 1

$\textbf{Given:}$

$\text{P is the midpoint of A(-10,4) and B(-2,0) and}$

$\text{P lies on the line joining C(-9-4) and D(-4,y)}$

$\textbf{To find:}$

$\text{The value of y and the ratio in which P divides CD}$

$\textbf{Solution:}$

$\text{Since P is the midpoint of AB, we have}$

$\text{P is}\,(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2})$

$\text{P is}\;(\dfrac{-10+(-2)}{2},\dfrac{4+0}{2})$

$\text{P is}\;(\dfrac{-12}{2},\dfrac{4}{2})$

$\text{P is}\,(-6,2)$

$\text{Let P divides CD internally in the ratio m:n}$

$\text{Using section formula, we get}$

$(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n})=(-6,2)$

$(\dfrac{m(-4)+n(-9)}{m+n},\dfrac{my+n(-4)}{m+n})=(-6,2)$

$(\dfrac{-4m-9n}{m+n},\dfrac{my-4n}{m+n})=(-6,2)$

$\implies\,\dfrac{-4m-9n}{m+n}=-6$

$\implies\,-4m-9n=-6m-6n$

$\implies\,-4m+6m=-6n+9n$

$\implies\,2m=3n$

$\implies\,\dfrac{m}{n}=\dfrac{3}{2}$

$\implies\boxed{\bf\,m:n=3:2}$

$\text{and}$

$\dfrac{my-4n}{m+n}=2$

$\dfrac{(3)y-4(2)}{3+2}=2$

$\dfrac{3y-8}{5}=2$

$\implies\,3y-8=10$

$\implies\,3y=10+8$

$\implies\,3y=18$

$\implies\boxed{\bf\,y=6}$

$\textbf{Answer:}$

$\textbf{P divides CD internally in the ratio 3:2 and}$

$\textbf{the value of y is 6}$

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