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In what ratio of line x-y-2=0 divides the line segment joining (3,-1) and (8,9)?​

Sagot :

TheAnimeGirl
  • Let the given points ( 3 , -1 ) and ( 8 , 9 ) be A and B respectively. Let A ( 3 , - 1 ) be ( x₁ , y₁ ) and B ( 8 , 9 ) be ( x₂ , y₂ ). Let the point P ( x , y ) divides the line segment of joining points A ( 3 , -1 ) and ( 8 , 9 ) in the ratio m : n. Let m be m₁ and n be m₂ We know that :

[tex] \large{ \tt{❁ \: USING \: INTERNAL \: SECTION \: FORMULA: }}[/tex]

[tex] \large{ \bf{✾ \: P(x \:, y \: ) = ( \frac{m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m_{2}} \: ,\: \frac{m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m_{2}}) }}[/tex]

[tex] \large{ \bf{⟹ \: ( \frac{8m + 3n}{m + n} , \: \frac{9m -n}{m + n}) }}[/tex]

  • Since point P lies on the line x - y - 2 = 0 ,

[tex] \large{ \bf{ ⟼\frac{8m + 3n}{m + n} - \frac{9m - n}{m + n} - 2 = 0 }}[/tex]

[tex] \large{ \bf{⟼ \: \frac{8m + 3n - 9m + n}{m + n} - 2 = 0 }}[/tex]

[tex] \large{ \bf{⟼ \: \frac{4n - m}{ m + n} - 2 = 0 }}[/tex]

[tex] \large{⟼ \: \bf{ \frac{4n - m}{m + n }} = 2} [/tex]

[tex] \large{ \bf{⟼ \: 4n - m = 2m + 2n}}[/tex]

[tex] \large{ \bf{⟼ \: 4n -2 n = 2m + m}}[/tex]

[tex] \large{ \bf{⟼2n = 3m}}[/tex]

[tex] \large{ \bf{⟼ \: 3m = 2n}}[/tex]

[tex] \large{ \bf{⟼ \: \frac{m}{n} = \frac{2}{3} }}[/tex]

[tex] \boxed{ \large{ \bf{⟼ \: m : \: n = 2: \: }3}}[/tex]

  • Hence , The required ratio is 2 : 3 .

-Hope I helped! Let me know if you have any questions regarding my answer and also notify me , if you need any other help! :)

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