Question

he points which divides the line segment of points P(-1, 7) and (4, -3) in the ratio of 2:3 is​

Answer 1

Given : We've provided with two endpoints of line segement are ( -1 , 7 ) and ( 4 , -3 ) .

Need To Find : The point which devides the line segment in the ratio of 2 : 3 ?

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❍ Let's Consider the point which devides the line segment in the ratio of 2 : 3 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 = -1 ,
• y₁ = 7 ,
• x₂ = 4 ,
• y₂ = -3 &,
• m₁ : m₂ = 2 :3

$\\\\ \qquad :\implies \sf \: \: ( \: x \:,\:y\:) \: \: \:=\:\Bigg( \: \dfrac{ m_1 \: x_2 \:\: +\: m_2 \:x_1 \:}{m_2 \:+ \:m_1 \:} \:, \: \dfrac{ m_1 \: y_2 \:\: +\: m_2 \:y_1 \:}{m_2 \:+ \:m_1 \:} \:\Bigg) \\\\\\ \qquad :\implies \sf \: \: ( \: x \:,\:y\:) \: \: \:=\:\Bigg( \: \dfrac{ 2\: ( 4 ) \:\: +\: 3 \:( -1 ) \:}{ 3 \:+ \:2 \:} \:, \: \dfrac{ 2 \: ( -3 ) \:\: +\: 3 \: ( 7 ) \:}{ 3\:+ \:2 \:} \:\Bigg) \\\\\\ \qquad :\implies \sf \: \: ( \: x \:,\:y\:) \: \: \:=\:\Bigg( \: \dfrac{ 8 \:\: +\: ( -3 ) \:}{ 5 \:} \:, \: \dfrac{ \:\: (-6)\: +\: 21 \:}{ \:5 \:} \:\Bigg) \\\\\\ \qquad :\implies \sf \: \: ( \: x \:,\:y\:) \: \: \:=\:\Bigg( \: \dfrac{ 5 \:}{ 5 \:} \:, \: \dfrac{ 15 \:}{ 5 \:} \:\Bigg) \\\\\\ \qquad :\implies \underline {\boxed{\pmb{\frak{ \: \: ( \: x \:,\:y\:) \: \: \:=\: \: (\:1\: \:, \: 3\:) \:}}}}\:\:\bigstar \:$

$\qquad \therefore \:\underline {\sf Hence, \: \:The\: \:Point\: \:which \:\:devides\: \:the \:\:line\: \:segment\: \:in\: \: 2\: :\: 3 \:\:is \:\:\pmb{\bf \:(\:1\: ,\: 3 \:)\:}\:.\:}$

Answer 2

Step-by-step explanation:

Given: P(-1,7) and Q(4,-3)

To find: Find the point which divides the line segment in ratio 2:3.

Solution:

Tip: Section formula

If line segment by joining the points $P(x_1,y_1)$and $Q(x_2,y_2)$ is divided by the S(x,y) in m:n ratio,then coordinates of S are given by

$\boxed{\bold{\red{x = \frac{mx_1 + nx_2}{m + n} }}} \\ \\ \boxed{\bold{\green{y = \frac{my_1 + ny_2}{m + n}}}}$

Here,

Points are P(-1,7) and Q(4,-3) ,ratio is 2:3

apply the values in the formula

$x = \frac{2(4) + 3( - 1)}{2 + 3}$

$x = \frac{8 - 3}{5}$

$x = \frac{5}{5}$

$\bold{\red{x = 1 }}$

by the same way,find y

$y = \frac{3 (7)+ 2( - 3)}{3 + 2}$

$y = \frac{21 - 6}{5}$

$y = \frac{15}{5}$

$\bold{\green{y = 3 }}$

Coordinates of S are (1,3).

Final answer:

Coordinates of S are (1,3).

Hope it helps you.

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