If the distance of point \(B(2,1,-3)\) from the line passing through the point \(A(4,-2,2)\), and parallel to the vector \(\overset{⃗}{c} = -4\hat{i}-6\hat{j}-2\hat{k}\) is \(x\), then \(x^4+x^2+541 =\)
Show Hint
Move along the line direction until the plane is reached.
Step 1: Direct formula:
Distance along direction $\vec d$ is $\frac{|ax_1 + by_1 + cz_1 - k|}{|a l + b m + c n|}\times|\vec d|$ for plane $ax + by + cz = k$ and direction $(l, m, n)$.
Step 2: Compute:
Numerator: $|1 - 0 + 3 - 9| = 5$. Denominator: $1\cdot2 - 1\cdot3 - 1\cdot(-6) = 5$. Ratio 1. Magnitude of direction vector is 7. Distance = 7 (B).
Final Answer:
7 units.
\[ \boxed{7\text{ units}} \]