Question

Find the coordinates of the point which divides
internally the join of the points
(a) (8,9) and (-7,4) in the ratio 2:3
(b) (1, -2) and (4,7) in the ratio 1:2.​

Answer 1

Answer:

internally the join of the points

(a) (8,9) and (-7,4) in the ratio 2:3

(b) (1, -2) and (4,7) in the ratio 1:2.

Answer 2

⠀⠀⠀⠀a) Let the point P (x,y) divides internally join the points A (8,9) and B (-7,4) in ratio 2:3.

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

$\underline{\bigstar\:\boldsymbol{Using\:Section\:Formula\::}}$

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

$\sf Here \begin{cases} & \sf{x_1\:,\: x_2 = \bf{8\:,\:-7}} \\ & \sf{y_1\:,\:y_2 = \bf{9\:,\:4}} \\ & \sf{m_1\:,\:m_2 = \bf{2\:,\:3}} \end{cases}$

$\dag\;{\underline{\frak{Putting\:values\:in\;formula,}}}$

$:\implies\sf (x,y) = \bigg( \dfrac{2 \times -7 + 3 \times 8}{2 + 3}\:,\: \dfrac{2 \times 4 + 3 \times 9}{2 + 3} \bigg)\\ \\ \\ :\implies\sf (x,y) = \bigg( \dfrac{-14 + 24}{5}\:,\: \dfrac{8 + 27}{5} \bigg)\\ \\ \\ :\implies\sf (x,y) = \bigg( \dfrac{10}{5}\:,\: \dfrac{35}{5} \bigg)\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{(x,y) = (2\:,7)}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{The\: coordinates\:of\:the\:point\:is\: {\textsf{\textbf{(2,7)}}}.}}}$

⠀⠀⠀⠀b) Let the point P (x,y) divides internally join the points A (1,-2) and B (4,7) in ratio 1:2.

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

Again,

$\underline{\bigstar\:\boldsymbol{Using\:Section\:Formula\::}}$

$\sf Here \begin{cases} & \sf{x_1\:,\: x_2 = \bf{4\:,\:1}} \\ & \sf{y_1\:,\:y_2 = \bf{-2\:,\:7}} \\ & \sf{m_1\:,\:m_2 = \bf{1\:,\:2}} \end{cases}$

$:\implies\sf (x,y) = \bigg( \dfrac{1 \times 4 + 2 \times 1}{1 + 2}\:,\: \dfrac{1 \times 7 + 2 \times - 2}{1 + 2} \bigg)\\ \\ \\ :\implies\sf (x,y) = \bigg( \dfrac{4 + 2}{3}\:,\: \dfrac{7 - 4}{3} \bigg)\\ \\ \\ :\implies\sf (x,y) = \bigg( \dfrac{6}{3}\:,\: \dfrac{3}{3} \bigg)\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{(x,y) = (2\:,1)}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{The\: coordinates\:of\:the\:point\:is\: {\textsf{\textbf{(2,1)}}}.}}}$