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Find the equation of the locus of a moving point which divides the line segment joining the points $(-a,0)$ and $(0,-b)$ in the ratio of $4:5$ and making $a+b=-18$.

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closed as off-topic by Alex Provost, SchrodingersCat, Davide Giraudo, Leucippus, Pragabhava Dec 28 '15 at 18:36

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  • $\begingroup$ You should include your own thoughts and attempts so that you might receive constructive feedback and avoid having your question put on hold. $\endgroup$ – Em. Dec 28 '15 at 17:37
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Let the coordinates of the moving point be $P(h, k)$ which divides the line segment joining the points $(x_1, y_1)\equiv(-a, 0)$ & $(x_2, y_2)\equiv(0, -b)$ in a ratio $m:n::4:5$ then using division formula,
$$(h, k)\equiv \left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$$ $$(h, k)\equiv \left(-\frac{5a}{9}, -\frac{4b}{9}\right)$$ setting the corresponding values, one should get

$$(h, k)\equiv \left(\frac{4(0)+5(-a)}{4+5}, \frac{4(-b)+5(0)}{4+5}\right)$$ $$(h, k)\equiv \left(-\frac{5a}{9}, -\frac{4b}{9}\right)$$ comparing the corresponding coordinates, one should get $$h=-\frac{5a}{9}\implies a=-\frac{9h}{5}$$ & $$k=-\frac{4b}{9}\implies b=-\frac{9k}{4}$$ Now, setting the values of $a$ & $b$ in the given relation: $a+b=-18$, $$-\frac{9h}{5}-\frac{9k}{4}=-18$$ $$4h+5k=40$$ hence, replacing $h=x$ & $k=y$, the locus of the moving point $(h, k)$ is $$\color{red}{4x+5y=40}$$

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