Plane $2x + 3y + 4z = 5$ is rotated about the line where it cuts the $xy$-plane by an angle $\alpha$. In the new position the plane contains the point $(1, 1, 1)$. If the angle $\alpha = \cos^{-1}\sqrt{\frac{p}{q}}, (p$ is rational number in its simplest form$)$ then $q - 2p = $ __________.
Step-by-Step Solution
Key Concept: The locus of midpoints of lines joining two point sets lies on a fixed plane when the coefficients of parameters in the parametric representation vanish.
The plane passing through the midpoint of $PQ$ for all values of $\lambda$ and $\mu$ must satisfy the condition that coefficients of $\lambda$ and $\mu$ vanish independently. Setting up the plane equation $ax + by + cz = 1$ and substituting midpoint coordinates, we get three conditions: $3a + b + 2c = 0$, $a + 2b + 3c = 0$, and $3a + 5b + 4c = 2$. Solving these simultaneously gives $a = \frac{1}{9}$, $b = \frac{7}{9}$, $c = -\frac{5}{9}$, yielding the plane $x + 7y - 5z = 9$.
Correct Answer: I need to solve this step-by-step.
**Step 1: Find the line of intersection of the plane with the xy-plane.**
The plane is $2x + 3y + 4z = 5$.
On the xy-plane, $z = 0$, so: $2x + 3y = 5$
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