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Find the ratio in which line $y = x$ divides the line segment joining the points $(6, -3)$ and $(1, 6)$.
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Let the ratio be $k:1$
$x = \frac{k(1)+1(6)}{k+1} = \frac{k+6}{k+1}$, $y = \frac{k(6)+1(-3)}{k+1} = \frac{6k-3}{k+1}$
$P(x, y)$ lies on $y = x$
$\Rightarrow \frac{k+6}{k+1} = \frac{6k-3}{k+1}$
$\Rightarrow k+6 = 6k-3$
$\Rightarrow 5k = 9 \Rightarrow k = \frac{9}{5}$
Ratio is $9:5$
$x = \frac{k(1)+1(6)}{k+1} = \frac{k+6}{k+1}$, $y = \frac{k(6)+1(-3)}{k+1} = \frac{6k-3}{k+1}$
$P(x, y)$ lies on $y = x$
$\Rightarrow \frac{k+6}{k+1} = \frac{6k-3}{k+1}$
$\Rightarrow k+6 = 6k-3$
$\Rightarrow 5k = 9 \Rightarrow k = \frac{9}{5}$
Ratio is $9:5$