Consider the three planes
Question:

Consider the three planes

$P_{1}: 3 x+15 y+21 z=9$

$P_{2}: x-3 y-z=5$, and

$P_{3}: 2 x+10 y+14 z=5$

Then, which one of the following is true?

1. (1) $\mathrm{P}_{1}$ and $\mathrm{P}_{3}$ are parallel.

2. (2) $\mathrm{P}_{2}$ and $\mathrm{P}_{3}$ are parallel.

3. (3) $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ are parallel.

4. (4) $\mathrm{P}_{1}, \mathrm{P}_{2}$ and $\mathrm{P}_{3}$ all are parallel.

Correct Option: 1

Solution:

$P_{1}=x+5 y+7 z=3$

$P_{2}=x-3 y-z=5$

$P_{3}=x+5 y+7 z=5 / 2$

$\Rightarrow P_{1} \| P_{3}$