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If the intercept of a line between the coordinate axes is divided by the point (–5, 4) in the ratio 1:2, then find the equation of the line.

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. Let a and b be the intercepts on the given line

Coordinates of A and B are (a,0)and (0,b) respectiively.

Using the section formula we find the value of a and b

$\left ( x,y \right )=\left ( \frac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}, \frac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\right )$

$\left ( -5,4\right )=\left ( \frac{1*0+2*a}{1+2}, \frac{1*b+2*0}{1+2}\right )=\left (\frac{2a}{3},\frac{b}{3} \right )$

$-5=\frac{2a}{3}$ and $4=\frac{b}{3}$

$-15=2a$ and $b=12$

$a=-\frac{15}{2}$ and $b=12$

Coordinates of A and B are $\left ( -\frac{15}{2},0 \right )$ and $\left ( 0 , 12 \right )$

Equation of line AB $y-0 = \frac{12-0}{0-\left ( -\frac{15}{2} \right )}\left ( x- \left ( -\frac{15}{2} \right ) \right )$

$y = \frac{12}{\frac{15}{2}}\left ( x+\frac{15}{2} \right )$

$y =\frac{24}{15}\left ( x+\frac{15}{2} \right )$

$y =\frac{8}{5}x+15$

$5y=8x+60$

Hence, trequired equation is 8x-5y+60=0

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