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$X O Z$-plane divides the join of $(2,3,1)$ and $(6,7,1)$ in the ratio:
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Verified Answer
The correct answer is:
$-3: 7$
Let $X O Z$-plane divides the joining the point $(2,3,1)$ and $(6,7,1)$ into $m: n$, then the co-ordinate of $X O Z$-plane is
$\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}, \frac{m z_2+n z_1}{m+n}\right)$
$=\left(\frac{6 m+2 n}{m+n}, \frac{7 m+3 n}{m+n}, \frac{m+n}{m+n}\right)$
But on the $X O Z$-plane $y$-coordinate will be zero, then
$\frac{7 m+3 n}{m+n}=0 \Rightarrow \frac{m}{n}=-\frac{3}{7}$
$\therefore \quad m: n=-3: 7$
$\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}, \frac{m z_2+n z_1}{m+n}\right)$
$=\left(\frac{6 m+2 n}{m+n}, \frac{7 m+3 n}{m+n}, \frac{m+n}{m+n}\right)$
But on the $X O Z$-plane $y$-coordinate will be zero, then
$\frac{7 m+3 n}{m+n}=0 \Rightarrow \frac{m}{n}=-\frac{3}{7}$
$\therefore \quad m: n=-3: 7$
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