JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Three Dimensional Geometry

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 ?

Answer: (A)

Solution

Two planes

a_{1} x + b_{1} y + c_{1} z = d_{1}

and

a_{2} x + b_{2} y + c_{2} z = d_{2}

are parallel if their normal vectors are proportional:

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}

.

For

P_{1}

and

P_{3}

\frac{3}{2} = \frac{15}{10} = \frac{21}{14} = \frac{3}{2}

, so their normals are proportional.

Check if they are the same plane:

\frac{9}{5} = 1.8 \neq = \frac{3}{2} = 1.5

, so they are distinct parallel planes.

For

P_{1}

and

P_{2}

\frac{3}{1} \neq = \frac{15}{- 3}

, not parallel.

For

P_{2}

and

P_{3}

\frac{1}{2} \neq = \frac{- 3}{10}

, not parallel.

Therefore,

P_{1}

and

P_{3}

are parallel.

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Single correct option