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Given that $P(3,2,-4),\:Q(5,4,-6)\:\:and\:\:R(9,8,-10)$ are collinear. Find the ratio in which $Q$ divides $PR$

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  • Section formula: The coordinates of the point $C$ that divides the segment joining the points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ in the ratio $l:m$ internally is given by $C\big(\large\frac{lx_2+mx_1}{l+m},\frac{ly_2+my_1}{l+m},\frac{lz_2+mz_1}{l+m}\big)$
Given: $P(3,2,-4),\:Q(5,4,-6)\:\:and\:\:R(9,8,-10)$ are collinear.
Let the ratio in which $Q$ divides $PR$ be $ k:1$
$\therefore$ We know from section formula that the coordinates of $Q$ is given by
But given that $Q(5,4,-6)$
$\Rightarrow\:4k=2$ or $k=\large\frac{1}{2}$
$\therefore$ the required ratio is $\large\frac{1}{2}$$\::\:1=1:2$
answered May 6, 2014 by rvidyagovindarajan_1

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