Animated Solution for Mathematics - Three Dimensional Geometry: Let $P_1$ and $P_2$ be two planes given by $P_1:10x+15y+12z-60=0$, $P_2:-2x+5y+4z-20=0$. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_1$ and $P_2$ ?

We are given two planes in 3D space: $P_1:10x+15y+12z-60=0$ $P_2:-2x+5y+4z-20=0$

These planes intersect along a line $L$. For a line to serve as an edge of a tetrahedron where two faces lie on $P_1$ and $P_2$, the line must either lie entirely within one of the planes or connect a point on $P_1$ to a point on $P_2$.

To determine if a line is a valid edge, we represent it parametrically as:

By substituting these expressions into the plane equations, we solve for the parameter $t$. - If we obtain a unique value for $t$, the line pierces the plane at a single point. - If we obtain the identity $0=0$, the line lies entirely within the plane. - If the line intersects both planes at the same point on $L$, it fails to form a distinct edge of the tetrahedron.

Option A and Option B: Upon substituting the parametric forms into the equations for $P_1$ and $P_2$, we find distinct values for $t$. This confirms that these lines bridge the two planes at different points, successfully creating a valid skew edge for the tetrahedron.

Option C: When we substitute the parametric form of this line into both plane equations, we arrive at the same value of $t$. This indicates that the line intersects both planes at a single point located on the intersection line $L$. Consequently, this line cannot serve as an edge of the tetrahedron.

Option D: Substituting the coordinates of this line into the equation for $P_2$ yields the identity $0=0$. This confirms that every point on the line satisfies the equation of $P_2$. Because the line lies entirely within the plane, it is a perfectly valid edge.

Through this geometric analysis, we have demonstrated that a line can serve as an edge if it is contained within one of the planes or if it connects two distinct points on the respective planes. Lines that intersect the planes at a single point on the line of intersection $L$ are excluded from being edges of the tetrahedron.