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Question Number 25441 by rather ishfaq last updated on 10/Dec/17
in what ratio in which y−x+2=0 divides the line joining (3,−1) and (8,9).
$${in}\:{what}\:{ratio}\:{in}\:{which}\:{y}−{x}+\mathrm{2}=\mathrm{0}\:{divides}\:{the}\:{line}\:{joining}\:\left(\mathrm{3},−\mathrm{1}\right)\:{and}\:\left(\mathrm{8},\mathrm{9}\right). \\ $$$$ \\ $$
Answered by mrW1 last updated on 10/Dec/17
Line from A(3,−1) to B(8,9):  (x,y)=(3,−1)+λ(5,10)  For intersection point C we have  (−1+10λ)−(3+5λ)+2=0  −4+5λ+2=0  ⇒λ=(2/5)  ⇒AC=(2/5)AB  that means the line AB is divided  in a ratio of 2:3 by the point C.
$${Line}\:{from}\:{A}\left(\mathrm{3},−\mathrm{1}\right)\:{to}\:{B}\left(\mathrm{8},\mathrm{9}\right): \\ $$$$\left({x},{y}\right)=\left(\mathrm{3},−\mathrm{1}\right)+\lambda\left(\mathrm{5},\mathrm{10}\right) \\ $$$${For}\:{intersection}\:{point}\:{C}\:{we}\:{have} \\ $$$$\left(−\mathrm{1}+\mathrm{10}\lambda\right)−\left(\mathrm{3}+\mathrm{5}\lambda\right)+\mathrm{2}=\mathrm{0} \\ $$$$−\mathrm{4}+\mathrm{5}\lambda+\mathrm{2}=\mathrm{0} \\ $$$$\Rightarrow\lambda=\frac{\mathrm{2}}{\mathrm{5}} \\ $$$$\Rightarrow{AC}=\frac{\mathrm{2}}{\mathrm{5}}{AB} \\ $$$${that}\:{means}\:{the}\:{line}\:{AB}\:{is}\:{divided} \\ $$$${in}\:{a}\:{ratio}\:{of}\:\mathrm{2}:\mathrm{3}\:{by}\:{the}\:{point}\:{C}. \\ $$

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