Question

Find the ratio in which the line segment joining the points (2,4,5) and (3,5,-4) is divided by the XY-plane ?​

Answer 1

$\underline{\textsf{GIVEN:}}$

$\textsf{Points are (2,4,5) and (3,5,-4)}$

$\underline{\textsf{TO FIND:}}$

$\textsf{The ratio in which the line segment joining the given points is divided by xy plane}$

$\underline{\textsf{SOLUTION:}}$

$\underline{\textsf{CONCEPT USED:}}$

$\boxed{\begin{minipage}{9cm} \\\;\;\textsf{The co ordinates of the point which divides the line segment}\\\\\mathsf{joining\;(x_1,y_1,z_1)\;and\;(x_2,y_2,z_2)\;internally\;in\;the\;ratio \;m:n}\\\\\mathsf{are\;\;\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n},\dfrac{mz_2+nz_1}{m+n}\right)}\\ \end{minipage}}$

$\textsf{Let P be the point which divides the line segment joining (2,4,5)}$

$\textsf{and (3,5,-4) interrnally in the ratio m:n)}$

$\textsf{Then, co-ordinates of P are}$

$\mathsf{\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n},\dfrac{mz_2+nz_1}{m+n}\right)}$

$\mathsf{\left(\dfrac{m(3)+n(2)}{m+n},\dfrac{m(5)+n(4)}{m+n},\dfrac{m(-4)+n(5)}{m+n}\right)}$

$\mathsf{\left(\dfrac{3m+2n}{m+n},\dfrac{5m+4n}{m+n},\dfrac{-4m+5n}{m+n}\right)}$

$\textsf{Since the line segment joining the given points is divided by xy plane,}$

$\textsf{z component of P is zero}$

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

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

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

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

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

$\underline{\textsf{FIND MORE:}}$

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