MHT CET · Maths · Three Dimensional Geometry

Equations of planes parallel to the plane $x - 2 y + 2 z + 4 = 0$ which are at a distance of one unit from the point $(1,2, 3)$ are …..

A $x + 2 y + 2 z = - 6, x + 2 y + 2 z = 5$
B $x - 2 y - 6 = 0, x - 2 y + z = 6$
C $x - 2 y + 2 z = 6, x + 2 y + 2 z = 0$
D $x - 2 y + 2 z = 0, x - 2 y + 2 z - 6 = 0$

Verified Solution

Answer & Solution

Correct Answer

(D) $x - 2 y + 2 z = 0, x - 2 y + 2 z - 6 = 0$

Step-by-step Solution

Detailed explanation

Given equation of plane

x - 2 y + 2 z + 4 = 0

….(i)
Equation of plane parallel (ii) from point (1, 2, 3) is 1 unit

∴ \frac{|1 (1) - 2 (2) + 2 (3) + k|}{\sqrt{{(1)}^{2} + {(- 2)}^{2} + {(2)}^{2}}} = 1

\Rightarrow \frac{|1 - 4 + 6 + k|}{\sqrt{1 + 4 + 4}} = 1

\Rightarrow \frac{|3 + k|}{\sqrt{9}} = 1

\Rightarrow |3 + k| = 3 \Rightarrow 3 + k = \pm 3

∴

x - 2 y + 2 z + 0 = 0

and

x - 2 y + 2 z - 6 = 0

\Rightarrow x - 2 y + 2 z = 0

and

x - 2 y + 2 z - 6 = 0

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Equations of planes parallel to the plane x-2y+2z+4=0 which are at a distance of one unit from the point 1,2,3 are …..

Step-by-step Solution

Equations of planes parallel to the plane $x - 2 y + 2 z + 4 = 0$ which are at a distance of one unit from the point $(1,2, 3)$ are …..