Question

3) The coordinates of the point which divides the join of (3,2) & (0,5)
in the ratio 2:1 are
A) (1,4) B)(4,1) C) (3, 7) D) (7,3)​

Answer 1

Given that , We've provided with the co – ordinates of the end point of line segment (3,2) & (0,5) .

Need To Find : The co – ordinates which devides the line segment in ratio 2 : 1 .

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━⠀

❍ Let's Consider the point which devides the line segment in the ratio of 2 : 1 be ( x , y ) .

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

$\qquad \:\star \:\:\underline{\boxed{\pmb{\sf { \: \: Section \: \: \:=\:\Bigg\lgroup \: \dfrac{ m_1 \: x_2 \:\: +\: m_2 \:x_1 \:}{m_2 \:+ \:m_1 \:} \:, \: \dfrac{ m_1 \: y_2 \:\: +\: m_2 \:y_1 \:}{m_2 \:+ \:m_1 \:} \:\Bigg\rgroup \:}}}}$

Where ,

• x₁ = 3 ,

• y₁ = 2 ,

• x₂ = 0 ,

• y₂ = 5 &,

• m₁ : m₂ = 2 : 1

$\\\\ \qquad :\implies \sf \: \: ( \: x \:,\:y\:) \: \: \:=\:\Bigg( \: \dfrac{ m_1 \: x_2 \:\: +\: m_2 \:x_1 \:}{m_2 \:+ \:m_1 \:} \:, \: \dfrac{ m_1 \: y_2 \:\: +\: m_2 \:y_1 \:}{m_2 \:+ \:m_1 \:} \:\Bigg) \\\\\\ \qquad :\implies \sf \: \: ( \: x \:,\:y\:) \: \: \:=\:\Bigg( \: \dfrac{ 2\: ( 0 ) \:\: +\: 1 \:( 3 ) \:}{ 1 \:+ \:2 \:} \:, \: \dfrac{ 2 \: ( 5 ) \:\: +\: 1 \: ( 2 ) \:}{ 1\:+ \:2 \:} \:\Bigg) \\\\\\ \qquad :\implies \sf \: \: ( \: x \:,\:y\:) \: \: \:=\:\Bigg( \: \dfrac{ 0 \:\: +\: 3 \:}{ 3 \:} \:, \: \dfrac{ 10 \:\: +\: 2 \:}{ 3 \:} \:\Bigg) \\\\\\ \qquad :\implies \sf \: \: ( \: x \:,\:y\:) \: \: \:=\:\Bigg( \: \dfrac{ \: 3 \:}{ 3 \:} \:, \: \dfrac{ \: 12 \:}{ 3 \:} \:\Bigg) \\\\\\ \qquad :\implies \sf \: \: ( \: x \:,\:y\:) \: \: \:=\:\Bigg( \: \cancel {\dfrac{ \: 3 \:}{ 3 \:} }\:, \: \cancel{ \dfrac{ \: 12 \:}{ 3 \:} }\:\Bigg) \\\\\\ \qquad :\implies \underline {\boxed{\pmb{\frak{ \: \: ( \: x \:,\:y\:) \: \: \:=\: \: (\:1\: \:, \: 4\:) \:}}}}\:\:\bigstar$

$\\\therefore \:\underline {\sf Hence, \:The\: Point\: which \:devides\: the \:line\: \:segment\: \:in\:ratio \: 2\: :\: 1 \:\:is \:\:\pmb{\sf \:Option \:A\:) \:(\:1\: ,\: 4 \:)\:}\:.\:}$