Question

Consider the three planes

$$\begin{gathered} P_1:3x+15y+21z=9 \\ {P}_2:x-3y-z=5 \\ {P}_{3:}2x+10y+14z=5 \end{gathered}$$

• A. $P_1$ and $P_3$ are parallel
• B. $P_2$ and $P_3$ are parallel
• C. $P_1$ and $P_2$ are parallel
• D. $P_1$, $P_2$ and $P_3$ all are parallel.

Answer 1

The correct option is A

$P_1$ and $P_3$ are parallel

Explanation for the correct option:

Option A:

Two planes are said to be parallel if the coefficients of the corresponding variables are in proportion or equal.

$$\begin{gathered} P_1:3x+15y+21z=9 \\ {P}_{3:}2x+10y+14z=5 \end{gathered}$$

Dividing equation $P_1$by $3.$

$P_1:x+5y+7z=3$

Dividing equation $P_2$by $2.$

$P_{3:}x+5y+7z=\frac{5}{2}$

The coefficients of $x,y,z$ are the same in both cases.

Hence, the correct option is A.

Explanation for the wrong option:

Option B:

$$\begin{gathered} P_2:x-3y-z=5 \\ {P}_{3:}2x+10y+14z=5 \end{gathered}$$

The coefficients of $x,y,z$ are not the same in both cases after simplification.

Option C:

$$\begin{gathered} P_1:3x+15y+21z=9 \\ {P}_2:x-3y-z=5 \end{gathered}$$

The coefficients of $x,y,z$ are not the same in both cases after simplification.

Option D:

$$\begin{gathered} P_1:3x+15y+21z=9 \\ {P}_2:x-3y-z=5 \\ {P}_{3:}2x+10y+14z=5 \end{gathered}$$

The coefficients of $x,y,z$ in $P_1$ and $P_3$ are the same. but the coefficients of $x,y,z$ in $P_2$ is different.

Hence, the correct option is A.