MHT CET · Maths · Three Dimensional Geometry

The co-ordinates of the foot of the perpendicular from the point \((0,2,3)\) on the line
\(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\) are

A \((-2,-3,1)\)
B \((2,3,-1)\)
C \((2,3,1)\)
D (-2,-3,-1)

Verified Solution

Answer & Solution

Correct Answer

(B) \((2,3,-1)\)

Step-by-step Solution

Detailed explanation

Let \(\alpha\) be the foot of the Ler drawn from the point \(\mathrm{P}(0,2,3)\) to the given line.
Co-ordinates of any point on given line are \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}=\lambda\)...say
Let \(Q \equiv(5 \lambda-3,2 \lambda+1,3 \lambda-4)\) and \(P \equiv(0,2,3)\)
\(\therefore\) d.r. of PQ are \(5 \lambda-3,2 \lambda-1,3 \lambda-7\)
d.r. of given line are \(5,2,3\)
\(\therefore 5(5 \lambda-3)+2(2 \lambda-1)+3(3 \lambda-7)=0\)
\(\Rightarrow 25 \lambda-15+4 \lambda-2+9 \lambda-21=0 \Rightarrow 38 \lambda-38=\) \(0 \Rightarrow \lambda=1\)
\(\therefore \mathrm{Q} \equiv(2,1,-4)\)

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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The co-ordinates of the foot of the perpendicular from the point \((0,2,3)\) on the line\(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\) are

Step-by-step Solution

The co-ordinates of the foot of the perpendicular from the point \((0,2,3)\) on the line
\(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\) are