Question

find the ratio in with the point P (k 7) divides the joint of A (8 9) and B (1 2) find k also​

Answer 1

$\textbf{Given:}$

$\textsf{Given points are A(8,9) and B(1,2)}$

$\textbf{To find:}$

$\textsf{The ratio in which the point P(k,7) divides the joining of A and B}$

$\textbf{Solution:}$

$\textbf{Section formula:}$

$\textsf{The co ordinates of the point which divides the}$

$\mathsf{line\;segment\;joining\;(x_1,y_1)\;and\;(x_2,y_2)\;internally\;in\;the\;ratio \;m:n\;are}$

$\boxed{\mathsf{\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)}}$

$\textsf{Let the point P divides AB internally in the ratio m:n}$

$\textsf{By section formula}$

$\textsf{The coordinates P are}$

$\mathsf{\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)}$

$\mathsf{\left(\dfrac{m(1)+n(8)}{m+n},\dfrac{m(2)+n(9)}{m+n}\right)}$

$\mathsf{\left(\dfrac{m+8n}{m+n},\dfrac{2m+9n}{m+n}\right)}$

$\mathsf{But,\;P\;is(k,7)}$

$\implies\mathsf{\left(\dfrac{m+8n}{m+n},\dfrac{2m+9n}{m+n}\right)=(k,7)}$

$\textsf{Equating corresponding coordinates on bothsides, we get}$

$\mathsf{\dfrac{m+8n}{m+n}=k\;\;\&\;\;\dfrac{2m+9n}{m+n}=7}$

$\mathsf{\dfrac{2m+9n}{m+n}=7}$

$\mathsf{2m+9n=7(m+n)}$

$\mathsf{2m+9n=7m+7n}$

$\mathsf{2m-7m=7n-9n}$

$\mathsf{-5m=-2n}$

$\mathsf{\dfrac{m}{n}=\dfrac{2}{5}}$

$\implies\boxed{\mathsf{m:n=2:5}}$

$\textsf{Also,}$

$\mathsf{\dfrac{m+8n}{m+n}=k}$

$\implies\mathsf{k=\dfrac{2+8(5)}{2+5}}$

$\implies\mathsf{k=\dfrac{2+40}{7}}$

$\implies\mathsf{k=\dfrac{42}{7}}$

$\implies\boxed{\mathsf{k=6}}$

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