Question

9.
The ratio in which the line segment joining A(3,-5) and B(5, 4) is divided by x-axis is :
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Answer 1

$\underline{\textbf{Given:}}$

$\textsf{A(3,-5) and B(5,4)}$

$\underline{\textbf{To find:}}$

$\textsf{The ratio in which line segment joining A(3,-5) and B(5,4)}$

$\textsf{is divided by x-axis}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Section formula:}}$

$\boxed{\begin{minipage}{9cm} \\The\;co\,ordintes\;of\;the\;point\;which\;divides\;the\;line\\\\segment\;joining\;(x_1,y_1)\;and\;(x_2,y_2)\;internally\;in\\\\the\;ratio\;m:n\;are\;\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)\\ \end{minipage}}$

$\textsf{Let the x-axis divides the line segment joining A(3,-5) and B(5,4) be m:n}$

$\textsf{By section formula}$

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

$\mathsf{\left(\dfrac{m(5)+n(3)}{m+n},\dfrac{m(4)+n(-5)}{m+n}\right)}$

$\mathsf{\left(\dfrac{5m+3n}{m+n},\dfrac{4m-5n}{m+n}\right)}$

$\textsf{Since it lies on x-axis, its y co-ordinate is zero}$

$\mathsf{\dfrac{4m-5n}{m+n}=0}$

$\mathsf{4m-5n=0}$

$\mathsf{4m=5n}$

$\implies\mathsf{\dfrac{m}{n}=\dfrac{5}{4}}$

$\implies\boxed{\mathsf{m:n=5:4}}$