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Question

If $$A$$ and $$B$$ are $$(-2, -2)$$ and $$(2, -4)$$, respectively, find the coordinates of $$P$$ such that $$AP$$ $$\displaystyle =\frac { 3 }{ 7 } AB$$ and $$P$$ lies on the line segment $$AB$$.


Solution

As given the coordinates of $$A (-2,-2 )$$ and $$B(2,-4)$$ and $$P$$ is a point lies on $$AB$$.

And $$AP=\dfrac{3}{7}AB$$

$$\therefore BP=\dfrac{4}{7}$$

Then, ratio  of $$AP$$ and $$PB=m_1:m_2=3:4$$

Let the coordinates of $$P$$ be $$(x,y)$$.

$$\therefore x=\dfrac{m_1x_2+m_2x_1}{m_1+m_2}$$

$$\Rightarrow x=\dfrac{3\times 2+4\times (-2)}{3+4}=\dfrac{6-8}{7}=\dfrac{-2}{7}$$

And $$ y=\dfrac{m_1y_2+m_2y_1}{m_1+m_2}$$

$$\Rightarrow y=\dfrac{3\times (-4)+4\times (-2)}{3+4}=\dfrac{-12-8}{7}=\dfrac{-20}{7}$$

$$\therefore $$Coordinates of $$P=$$$$\dfrac{-2}{7},\dfrac{-20}{7}$$

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Mathematics

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