Question

if point P ( -4 , 6) divides the line segment AB with A ( -6 , 10) in the ratio 2:1 , then coordinates of point B are ---

Answer 1

Given : Point P ( -4 , 6) divides the line segment AB with A ( -6 , 10) in the ratio 2 : 1 .

Need To Find : The co — ordinantes of Point B ?

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀

❍ Let's say , that the co — ordinantes of point B be ( x₂ , y₂ ) .

⠀⠀☯︎⠀We know that , the co – ordinantes of the point P( x , y ) Dividing the line segment joining the two points A( x₁ , y₁ ) and B( x₂ , y₂ ) , in the ratio of m₁ : m₂ is Section Formula and It's given by —

$\qquad \star\:\underline {\boxed {{\sf \:(x,y)\:=\: \Bigg( \dfrac{m_1x_2 + m_2x_1 }{ m_1+ m_2 }\:,\: \dfrac{m_1y_2 + m_2y_1 }{ m_1+ m_2 }\:\Bigg)\:\:}}}$

$\:\:\pmb{\sf Where \:}\begin {cases}\:\quad \sf (x_1,y_1)\:=\:\frak{ (-6,10)}\:\\\\ \:\quad \sf m_1:m_2\:=\:\frak{2:1}\\\\\:\quad\sf (x,y)\:=\:\frak{ (-4,6)}\:\end{cases}$

$\qquad \dag\underline {\frak{\:Substituting \:Known \:Values\:in\:Formula \:\::\:}}$

$:\implies \sf \:(x,y)\:=\: \Bigg( \dfrac{m_1x_2 + m_2x_1 }{ m_1+ m_2 }\:,\: \dfrac{m_1y_2 + m_2y_1 }{ m_1+ m_2 }\:\Bigg)\:\:$

$:\implies \sf \:(-4,6)\:=\: \Bigg( \dfrac{2(x_2) + 1(-6) }{ 2 + 1 }\:,\: \dfrac{2(y_2) + 1(10) }{ 2 + 1 }\:\Bigg)\:\:$

$:\implies \sf \:(-4,6)\:=\: \Bigg( \dfrac{2(x_2) - 6 }{ 2 + 1 }\:,\: \dfrac{2(y_2) + 10 }{ 2 + 1 }\:\Bigg)\:\:$

$:\implies \sf \:(-4,6)\:=\: \Bigg( \dfrac{2(x_2) - 6 }{ 3 }\:,\: \dfrac{2(y_2) + 10 }{ 3 }\:\Bigg)\:\:$

$:\implies \sf \: \dfrac{2(x_2) - 6 }{ 3 }\:=\:-4\:,\: \dfrac{2(y_2) + 10 }{ 3 }\:=\:6\:\:\:$

$:\implies \sf \: 2(x_2) - 6 \:=\:-12\:,\: 2(y_2) + 10 \:=\:18\:\:\:$

$:\implies \sf \: 2(x_2) \:=\:-12\:+\:6\:,\: 2(y_2) \:=\:18 - 10\:\:\:$

$:\implies \sf \: 2(x_2) \:=\:-6\:,\: 2(y_2) \:=\:8\:\:\:$

$:\implies \sf \: x_2 \:=\:\dfrac{-6}{2}\:,\: y_2 \:=\:\dfrac{8}{2}\:\:\:$

$:\implies \sf \: x_2 \:=\:-3\:,\: y_2 \:=\:4\:\:$

$:\implies \pmb {\underline {\boxed {\purple {\:\frak{ (\:-3\:,\:4\:)\:}}}}}\:\bigstar \:$

∴ Hence, co — ordinantes of B are ( – 3 , 4 ) .