Let $\mathrm{P}(\alpha, 4,7)$ and $\mathrm{Q}(3, \beta, 8)$ are two points. If YZ - plane divides the join…
Let $\mathrm{P}(\alpha, 4,7)$ and $\mathrm{Q}(3, \beta, 8)$ are two points. If YZ - plane divides the join of the points $P$ and $Q$ in the ratio 2:3 and ZX - plane divides the join of P and Q in the ratio $4: 5$. then length of line segment PQ is
$\sqrt{107}$
$\sqrt{27}$
$\sqrt{83}$
$\sqrt{97}$
Solution
Given, $P(\alpha, 4,7)$ and $Q(3, \beta, 8)$
Since, YZ - plane divide the line joining the point P and Q in ratio $=2: 3$
So, its x -coordinate is zero
$\Rightarrow \frac{3 \alpha+2.3}{2+3}=0 \Rightarrow \alpha=-2$
and ZX -plane divide the line joining the point $P$ and $Q$ in ratio $4: 5$
So, its y-coordinate is zero
$\Rightarrow \frac{20+4 \beta}{4+5}=0 \Rightarrow \beta=-5$
Now, $\mathrm{P}(-2,4,7)$ and $\mathrm{Q}(3,-5,8)$
So, $P Q=\sqrt{5^2+(-9)^2+1^2}=\sqrt{107}$