Geometry

# 3D Coordinate Geometry - Equation of a Line

If the line represented by $\frac{x-2}{2} = \frac{y-3}{3} = \frac{z-12}{5}$ passes through the point $(4,$ $a,$ $b)$, what is the value of $a+b$?

A line $l$ is parallel to the planes $-4x + y = 0$ and $x + 5z = 0.$ Given that it passes through the point $(4, 0, 6),$ find the equation of $l$ in the simplest form of $\frac{x-a}{p} = \frac{y-b}{q} = \frac{z-c}{r}.$

The following are the equations of lines that are parallel to one another: $\begin{cases} \frac{x-1}{2} = \frac{y-2}{p} = \frac{z-4}{p} _, \\\\ \frac{x-2}{q} = \frac{y-4}{2} = \frac{z-7}{3q-q^2}_, \\\\ \frac{x}{50} = \frac{y}{11 r - 10}=\frac{z}{r^2} _. \end{cases}$ Find the value of $p+q+r.$

The following is the equation of a line with direction vector $\vec{d} =(p,\text{ }q,\text{ }10):$ $\frac{2x-u}{4}=\frac{y-2}{2} =\frac{z-2}{5}.$ If the line passes through the point $(4,\text{ } 4,\text{ } 7),$ what is $p+q+u$?

If the two lines $\frac{x-4}{2} = \frac{y+3}{3} = \frac{z-4}{4} \text{ and } x-6 = \frac{y-k}{2} =z-3$ intersect, what is $k?$

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