Let a plane P has the points i^, j^ and i^+ j^+k^. Let L be the line through the point A3i^-5k^ and parallel to the vector i^- j^+k^, then equations of the normal to the plane P through A are

TS EAMCET · Maths · Three Dimensional Geometry

Let a plane $P$ has the points $\hat{i}, \hat{j}$ and $\hat{i} + \hat{j} + \hat{k}$ . Let $L$ be the line through the point $A$ $(3 \hat{i} - 5 \hat{k})$ and parallel to the vector $\hat{i} - \hat{j} + \hat{k}$ , then equations of the normal to the plane $P$ through $A$ are

A $\frac{x - 3}{1} = \frac{y}{1} = \frac{z - 5}{- 1}$
B $\frac{x + 3}{1} = \frac{y - 6}{1} = \frac{z - 1}{- 1}$
C $\frac{x + 3}{1} = \frac{y}{1} = \frac{z - 5}{- 1}$
D $\frac{x + 3}{1} = \frac{y - 6}{- 1} = \frac{z + 1}{1}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{x - 3}{1} = \frac{y}{1} = \frac{z - 5}{- 1}$

Step-by-step Solution

Detailed explanation

Given, A plane P has the points i^, j^ and i^+ j^+k^ Now let equation of plane be ax+by+cx+d=0, Now satisfying the given point one by one we get, a+d=0, b+d & a+b+c+d=0 ⇒a=-d, b=-d & c=d Now putting the value in equation we…

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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