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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