Question

The co-ordinates of the point P which divides the line joining the points (-14).(8,6) internally in the ratio 3:2 is​

Answer 1

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

$\sf \implies Coordinates \: of \: point \: A = (-1, 4)$

$\sf \implies Coordinates \: of \: point \: B = (8, 6)$

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

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

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

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

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

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

$\bf \underline{Where},$

$\sf \implies m_1 = 3$

$\sf \implies m_2 = 2$

$\sf \implies x_1 = - 1$

$\sf \implies x_2 = 8$

$\sf \implies y_1 = 4$

$\sf \implies y_2 = 6$

$\sf \underline{Substituting \: the \: values},$

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

$\bf \underline{Now},$

$\sf{(x,y) = \bigg( \dfrac{2 \times (- 1) + 3 \times 8}{3 + 2}\;,\; \dfrac{2 \times 4 + 3 \times 6}{3 + 2} \bigg)}$

$\sf{(x,y) = \bigg( \dfrac{ - 2 + 24}{5}\;,\; \dfrac{8 + 18}{5} \bigg)}$

$\sf{(x,y) = \bigg( \dfrac{22}{5}\;,\; \dfrac{26}{5} \bigg)}$

$\underline{\boxed{\bf \red{Hence , the \: required \: coordinates \: of \: the \: point \: are \bigg( \dfrac{ 22}{5} \:,\: \dfrac{26}{5}\bigg)}}}$

Answer 2

✿ Given" :-

• Coordinates of point A = (-1, 4)

• Coordinates of point B = (8, 6)

• Ratio in which P divides A and B is 3 : 2

✿ To find" :-

• Coordinates of point P = ?

✿ Solution" :-

$\: \: \: \: \maltese \: \sf \underline\red{Formula \: used \: by : }$

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

❒ Where as ,

$\: \: \: \: \: \: \: \: \: \: \: \: \sf \red{: \: \longmapsto} m_1 = 3 \\ \\ \sf \purple{: \: \longmapsto} m_2 = 2 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \red{: \: \longmapsto} x_1 = - 1 \\ \\ \sf \purple{: \: \longmapsto} x_2 = 8 \\ \\ \: \: \: \: \: \: \: \: \: \: \: \sf \red{: \: \longmapsto} y_1 = 4 \\ \\ \sf \purple{: \: \longmapsto} y_2 = 6$

❒ Substituting the values,

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

❒ And Next,

$\: \: \: \: \: \: \: \: \: \pink{: \implies}\rm{(x,y) = \bigg( \dfrac{2 \times (- 1) + 3 \times 8}{3 + 2}\;,\; \dfrac{2 \times 4 + 3 \times 6}{3 + 2} \bigg)} \\ \\ \pink{: \implies}\rm(x,y)=( 3+22×(−1)+3×8,3+22×4+3×6)$

$\maltese\rm{(x,y) = \bigg( \dfrac{ - 2 + 24}{5}\;,\; \dfrac{8 + 18}{5} \bigg)} \\ \\ \maltese \:\rm(x,y)=( 5−2+24, 58+18)$

$\purple\longmapsto\rm{(x,y) = \bigg( \dfrac{22}{5}\;,\; \dfrac{26}{5} \bigg)} \\ \\ \: \: \: \: \: \: \: \: \: \: \purple \longmapsto\rm(x,y)=( 522, 526)$

$\: \: \: \: \: \: ❍ \: {\underline{\sf\purple{Hence , the \: required \: coordinates \: of \: the \: point \: are \bigg( \dfrac{ 22}{5} \:,\: \dfrac{26}{5}\bigg)}}}$