Question

find the coordinates of the point which divides the line segment joining the point (2,-1) and (-3,4) internally in ratio 3:2 ​

Answer 1

$\bf \underline{ \underline{\maltese\:Given} }$

Coordinates of the point are A(2,-1) and B(-3,4) and the ratio of the line segment which is 3:2

Let P(x,y) be a point that divides the line joing the point A and B in the ratio 3:2 internally.

$\sf \implies Ratio \: in \: which \: P \: divides \: A \: and \: B \: is \: 3:2$

$\bf \underline{\underline{\maltese\: To \: find }}$

$\sf \implies Coordinates \: of \: point \:  P = \: ?$

$\bf \underline{\underline{\maltese\: Solution }}$

$\bf \underline{ Using\;section\;formula} :$

${\boxed{\sf{(x,y) = \bigg( \dfrac{m_2 x_1 + m_1 x_2}{m_1 + m_2}\;,\; \dfrac{m_2 y_1 + m_1 y_2}{m_1 + m_2} \bigg)}}}$

$\bf \underline{Where},$

$\sf \implies m_1 = 3$

$\sf \implies m_2 = 2$

$\sf \implies x_1 = 2$

$\sf \implies x_2 = - 3$

$\sf \implies y_1 = - 1$

$\sf \implies y_2 = 4$

$\bf \underline{Now, putting \: the \: values},$

$\sf{(x,y) = \bigg( \dfrac{2 \times 2 + 3 \times ( - 3)}{3 + 2}\;,\; \dfrac{2 \times (- 1) + 3 \times 4}{3 + 2} \bigg)}$

$\sf{(x,y) = \bigg( \dfrac{4 + (- 9)}{3 + 2}\;,\; \dfrac{ - 2 +12}{3 + 2} \bigg)}$

$\sf{(x,y) = \bigg( \dfrac{4 - 9}{5}\;,\; \dfrac{ - 2 +12}{5} \bigg)}$

$\sf{(x,y) = \bigg( \dfrac{- 5}{5}\;,\; \dfrac{10}{5} \bigg)}$

$\sf{(x,y) = \bigg( - 1\;,\; 2 \bigg)}$

$\underline{\boxed{\bf \red{Hence , the \: required \: coordinates \: of \: the \: point \: are \bigg( - 1 \:,\: 2\bigg)}}}$

Answer 2

❍ Let P(x,y) be the point which divides the line segment internally.

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★ Using the section formula for the internal division, i.e. -

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• ${\underline{\underline{\sf{\red{(x,y) = \bigg( \dfrac{m_2 x_1 + m_1 x_2}{m_1 + m_2}\;,\; \dfrac{m_2 y_1 + m_1 y_2}{m_1 + m_2} \bigg)}}}}}$

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

• $\sf{m_1\:=\:3}$
• $\sf{m_2\:=\:2}$
• $\sf{(x_1,\:y_1)\:=\:(2,\:-1)}$
• $\sf{(x_2,\:y_2)\:=\:(-3,\:4)}$

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Putting the above values in the above formula, we get :

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$\:\:\:\::\:\Longrightarrow\:\sf{(x,y)\:=\:{\dfrac{3\:(-3)\:+\:2\:(2)}{3\:+\:2}}\:\:\:,\:\:\:{\dfrac{3\:(4)\:+\:2\:(-1)}{3\:+\:2}}}$

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$\:\:\:\::\:\Longrightarrow\:\sf{(x,y)\:=\:{\dfrac{-9\:+\:4}{5}}\:\:\:,\:\:\:{\dfrac{12\:-\:2}{5}}}$

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$\:\:\:\::\:\Longrightarrow\:\sf{(x,y)\:=\:{\dfrac{-5}{5}}\:\:\:,\:\:\:{\dfrac{10}{5}}}$

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$\:\:\:\::\:\Longrightarrow\:\sf{(x,y)\:=\:-1\:\:\:,\:\:\:2}$

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$\:\therefore\:{\underline{\sf{Hence,\:{\textsf{\textbf{(-1,\:2)}}}\:is\: the \:point \:which \:divides\: the\: line \:segment \:internally}}}.$

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