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Question

If point $$P(9a -2 , -b) $$ divides the line segment joining the points $$ A(3a + 1 , -3) $$ and $$ B(8a , 5) $$ in the ratio $$ 3 : 1 $$, then find the values of $$a$$ and $$b$$ . 


Solution

Using the section formula, if a point $$(x,y)$$ divides the line joining the points $$({ x }_{ 1 },{ y }_{ 1 })$$ and $$({ x }_{ 2 },{ y }_{ 2 })$$ in the ratio $$ m:n $$, then 

$$(x,y) = \left( \dfrac { m{ x }_{ 2 } + n{ x }_{ 1 } }{ m + n },\dfrac { m{ y }_{ 2 }  + n{ y }_{ 1 } }{ m + n }  \right) $$
Point $$ P(9a - 2 , -b) $$ divides the line segment joining the points $$ A(3a + 1 , -3) $$ and $$ B(8a , 5) $$ in the ratio $$ 3 : 1 $$ . But , the coordinates of $$ P $$ are $$ (9a - 2 , -b ) $$ . 
Using section formula , we have 

$$ 9a - 2 = \dfrac{3(8a) + 1 (3a + 1)}{3 + 1} $$ 

$$ = \dfrac{24a + 3a + 1}{4} $$ 

$$ \Rightarrow \,36 a - 8 = 27 a + 1 $$ 
$$ \Rightarrow \,36a - 27a = 8 + 1 $$ 
$$ \Rightarrow \,9a = 9 $$ 
$$ \Rightarrow \,a = \dfrac{9}{9} = 1 $$ 
And
$$ -b = \dfrac{3(+5) + 1(-3)}{3 + 1} $$ 

$$ \Rightarrow \,-b = \dfrac{+15 - 3}{4} = \dfrac{12}{4} $$ 

$$ \Rightarrow \,b = -3 $$ 
Hence , $$ a = + 1 $$ and $$ b = -3 $$  

Mathematics

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