Geometry

# 3D Coordinate Geometry - Problem Solving

Determine the value of $b$ such that the line $\frac{x-1}{4} = \frac{y-2}{b} = \frac{z}{9}$ does not intersect the plane $2x- 4y + 5z = 6.$

If the angle between the line $x= \frac{y-4}{2}=\frac{z-1}{\lambda}$ and the plane $x + 3y + z = 1$ is $\cos ^{ -1 }{ \left( \sqrt { \frac { 10 }{ 11 } } \right) },$ what is $\lambda?$

Find the symmetric point of $A = (4, 4, 16)$ to the plane $\pi : x+ y+z=21.$

Let $\alpha,\beta,$ and $\gamma$ be the angles that the line $\frac{x-2}{3}=-\frac{y+1}{4}=\frac{z+3}{5}$ forms with the $xy$-, $yz$-, and $xz$-planes. Then what is the value of $\sin^2\alpha+\sin^2\beta+\sin^2\gamma?$

What is the volume of the region bounded by the plane $7x+8y+9z-504=0$ and the $xy$-, $yz$-, and $xz$-planes?

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