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Question

Find the coordinates of the points which divide the line segment joining $$ A( -2, 2)$$ and $$B(2, 8)$$ into four equal parts.


Solution

Let $$P$$,$$Q$$,$$R$$ be the points which divide the line into $$4$$ equal parts 

Then the ratio of $$AP$$ and $$PB=$$$$m_1,m_2=1:3$$

Here $$A(x_1,y_1)=A(-2,2),B(x_2,y_2)=B(2,8)$$

$$\therefore $$ Co-ordinates of P$$=\left(\dfrac{m_1x_2+m_2x_1}{m_1+m_2},\dfrac{m_1y_2+m_2y_1}{m_1+m_2} \right)$$

$$\Rightarrow \left(\dfrac{1\times 2-2\times 3}{1+3},\dfrac{1\times 8+3\times 2}{1+3} \right)$$

$$\Rightarrow \left(\dfrac{2-6}{4},\dfrac{8+6}{4} \right)=\left(-1,\dfrac{7}{2} \right)$$

The ratio of $$AQ$$ and $$QB$$=$$m_1:m_2=2:2$$

Here $$A(x_1,y_1)=A(-2,2),B(x_2,y_2)=B(2,8)$$

$$\therefore $$ Co-ordinates of Q$$=\left(\dfrac{m_1x_2+m_2x_1}{m_1+m_2},\dfrac{m_1y_2+m_2y_1}{m_1+m_2} \right)$$

$$\Rightarrow \left(\dfrac{2\times -2+2\times 2}{1+3},\dfrac{2\times 8+2\times 2}{1+3} \right)$$

$$\Rightarrow \left(\dfrac{-4+4}{4},\dfrac{16+4}{4} \right)=\left(0,5 \right)$$

Then the ratio of $$AR$$ and $$RB$$=$$m_1:m_2=3:1$$

Here $$A(x_1,y_1)=A(-2,2),B(x_2,y_2)=B(2,8)$$

$$\therefore $$ Co-ordinates of R$$=\left(\dfrac{m_1x_2+m_2x_1}{m_1+m_2},\dfrac{m_1y_2+m_2y_1}{m_1+m_2} \right)$$

$$\Rightarrow \left(\dfrac{3\times 2+1\times -2}{1+3},\dfrac{3\times 8+1\times 2}{1+3} \right)$$

$$\Rightarrow \left(\dfrac{6-2}{4},\dfrac{24+2}{4} \right)=\left(1,\dfrac{13}{2} \right)$$

$$\therefore$$ Coordinates are $$P(-1,\dfrac{7}{2}),(0,5),(1,\dfrac{13}{2})$$

Mathematics
RS Agarwal
Standard X

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