Question

find the ratio in which point p(1, 2) divides the join of A (-2, 1) and B (7, 4) ​

Answer 1

❒ DIAGRAM:

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$\setlength{\unitlength}{14mm}\begin{picture}(7,5)(0,0)\thicklines\put(0,0){\line(1,0){5}}\put(5.1, - 0.3){\sf B}\put( - 0.2, - 0.3){\sf A}\put(5.2, 0){\sf (7,4)}\put( - 0.7, 0){\sf (-2,1)}\put(2.3, 0.2){\sf P}\put(2.2, - 0.3){\sf (1,2)}\put(5, 0){\circle*{0.1}}\put(2.4, 0){\circle*{0.1}}\put(0, 0){\circle*{0.1}}\put(1,0.2){\sf k}\put(3.5, 0.2){\sf 1}\end{picture}$⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• NOTE: See the diagram from website only.

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Let the point P(1, 2) dividing the joining points A(-2, 1) and B(7, 4) in the Ratio of k:1 respectively.

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Now,

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$\underline{\bigstar\:\boldsymbol{Using\:section\:formula\::}}$⠀⠀

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$\dag\:\boxed{\sf{\pink{x = \dfrac{x_{1}m_{1} +x_{2}m_{1}}{m_{1} + m_{2}}}}}$⠀

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$\underline{\bf{\dag} \:\mathfrak{Substituting\: values \; now\: :}}$

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$:\implies\sf 1 = \dfrac{-2 (1) + 7 (k)}{k + 2} \\\\\\:\implies\sf k + 1 = - 2 + 7k \\\\\\:\implies\sf k - 7k = -2 - 1 \\\\\\:\implies\sf \cancel{-}\;6k = \cancel{-}\; 3 \\\\\\:\implies\sf 6k = 3k \\\\\\:\implies\sf k = \cancel\dfrac{3}{6} \\\\\\:\implies{\underline{\boxed{\frak{\pink{k = \dfrac{1}{2}}}}}}\;\bigstar$

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$\therefore\:{\underline{\sf{Hence, \ the \ required \ ratio \: \ is\: \bf{1:2}.}}}$

Answer 2

Answer:

Given :-

• A point P(1 , 2) divides the join of A(- 2 , 1) and B(7 , 4).

To Find :-

• What is the ratio.

Formula Used :-

${\red{\boxed{\large{\bold{x =\: \dfrac{x _ 1m_2 + x_2m_1}{m_1m_2}}}}}}$

Solution :-

Let, P(1 , 2) divides the join point of A(- 2 , 1) and B(7 , 4) in the ratio of k : 1.

According to the question by using the formula we get,

⇒ $\sf 1 =\: \dfrac{- 2 \times 1 + 7 \times k}{k + 1}$

By doing cross multiplication we get,

⇒ $\sf k + 1 =\: - 2 \times 1 + 7 \times k$

⇒ $\sf k + 1 = - 2 + 7k$

⇒ $\sf k - 7k = - 2 - 1$

⇒ $\sf - 6k = - 3$

⇒ $\sf k =\: \dfrac{\cancel{-}3}{\cancel{- }6}$

⇒ $\sf k =\: \dfrac{\cancel{3} \: ^{1}}{\cancel{6} \: ^{2}}$

➠ $\sf\bold{\purple{k =\: \dfrac{1}{2}}}$

$\therefore$ The ratio is 1 : 2 .