Question

The line segment LM is divided by point B (-7, 2) in the ratio 2 : 1. If L (5, 4), then find the co-ordinates of M.​

Answer 1

Given :

• Line segment LM is divided by a point B in ratio 2 : 1
• co-ordinates of point B are ( -7,2 )
• co-ordinates of point L are ( 5,4 )

To find :

• co-ordinates of M let ( p, q )

Knowledge required :

• Section formula

$\red{\bigstar}\boxed{\sf{(x,y)=\left(\dfrac{m\;x_2+n\;x_1}{m+n}\;,\;\dfrac{m\;y_2+n\;y_1}{m+n}\right)}}$

[ where (x,y) gives the coordinates of point diving a line segment AB in ratio m : n , such that coordinates of A are ( x₁, y₁ ) and coordinates of B are ( x₂, y₂ ). ]

Solution :

Let, the coordinates of point M be ( p, q )

then,

Using section formula

$\implies\sf{(-7,2)=\left(\dfrac{(2)\;(p)+(1)\;(5)}{(2)+(1)}\;,\;\dfrac{(2)\;(q)+(1)\;(4)}{(2)+(1)}\right)}$

$\implies\sf{(-7,2)=\left(\dfrac{2p+5}{3}\;,\;\dfrac{2q+4}{3}\right)}$

so,

$\implies\sf{\dfrac{2p+5}{3}=-7 \;\;\;and\;\;\;\dfrac{2q+4}{3}=2}$

$\implies\sf{2p+5=-21 \;\;\;and\;\;\;2q+4=6}$

$\implies\sf{2p=-26 \;\;\;and\;\;\;2q=2}$

$\red{\implies}\boxed{\sf{p=-13\;\;\;and\;\;\;q=1}}$

hence,

• ( p, q ) = ( -13, 1 )

therefore,

• Coordinates of point M are ( -13 , 1 ) .

Answer 2

$\huge\mathcal{\underbrace{\red{SOLUTION:-}}}$

GIVEN :-

• The line segment LM is divided by point B(-7 , 2) in the ratio ‘2 : 1’ .

• The coordinates of “L(5 , 4)” .

TO FIND :-

• The co-ordinates of M .

CALCULATION :-

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CONCEPT :-

✪ If the coordinates of the point P be (x , y), which divides the line segment joining $\rm{A\:(x_1\:,\:y_1)\:\:and\:\:B\:(x_2\:,\:y_2)\:}$ in the ratio ‘$\rm{(m_1:m_2)}$’, then the coordinates of the point ‘P’ is,

$\checkmark\:\rm{\red{\boxed{\purple{(x\:,\:y)\:=\:\left\{\:\dfrac{m_1\:x_2\:+\:m_2\:x_1}{m_1\:+\:m_2}\:,\:\dfrac{m_1\:y_2\:+\:m_2\:y_1}{m_1\:+\:m_2}\:\right\}\:}}}}$

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✍️ Let, the coordinates of the point M is (m , n) .

✍️ Here the point B(-7 , 2) which divides the line segment joining L(5 , 4) and M(m , n) in the ratio (2 : 1) . So, the formula for the coordinates of the point ‘B’ is,

$\rm{(-7\:,\:2)\:=\:\left\{\:\dfrac{2\times{m}\:+\:1\times{5}}{2\:+\:1}\:,\:\dfrac{2\times{n}\:+\:1\times{4}}{2\:+\:1}\:\right\}\:}$

$\rm{\implies\:(-7\:,\:2)\:=\:\left\{\:\dfrac{2m\:+\:5}{3}\:,\:\dfrac{2n\:+\:4}{3}\:\right\}\:}$

$\rm{\implies\:\dfrac{2m\:+\:5}{3}\:=\:-7\:\:and\:\:\dfrac{2n\:+\:5}{3}\:=\:2\:}$

$\rm{\implies\:2m\:+\:5\:=\:-7\times{3}\:\:and\:\:2n\:+\:4\:=\:2\times{3}\:}$

$\rm{\implies\:2m\:=\:-21\:-\:5\:\:and\:\:2n\:=\:6\:-\:4\:}$

$\rm{\implies\:m\:=\:\dfrac{-26}{2}\:\:and\:\:n\:=\:\dfrac{2}{2}\:}$

$\rm{\blue{\boxed{\implies\:m\:=\:-13\:\:and\:\:n\:=\:1\:}}}$

$\therefore\:\rm{(m\:,\:n)\:=\:(-13\:,\:1)\:}$

✍️ So, the coordinates of the point M is (-13 , 1) .