Animated Solution for Mathematics - Three Dimensional Geometry: Let the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lie in the plane $x+3y-\alpha z+\beta =0$. Then $(\alpha ,\beta )$ equals

To ensure the line defined by

lies entirely within the plane $x+3y-\alpha z+\beta =0$, we must satisfy two geometric constraints.

First, every point on the line must satisfy the equation of the plane. Second, the direction vector of the line must be perpendicular to the normal vector of the plane.

From the symmetric form of the line, we can identify a specific point $P$ that lies on the line: $P(2,1,-2)$.

Since the line is contained within the plane, the coordinates of $P$ must satisfy the plane equation $x+3y-\alpha z+\beta =0$. Substituting these values, we obtain:

Simplifying this expression leads to:

This yields our first master equation:

Next, we identify the direction vector of the line, $\vec{d}=(3,-5,2)$, and the normal vector of the plane, $\vec{n}=(1,3,-\alpha )$.

For the line to be parallel to the plane, the dot product of these two vectors must be zero: $\vec{d}\cdot \vec{n}=0$.

Calculating the dot product:

Solving for $\alpha$, we find:

With the value of $\alpha$ determined, we substitute it back into our first master equation to solve for $\beta$:

We have successfully determined the constants. The final values are $\alpha =-6$ and $\beta =7$.