Find equation for line parallel to line to and passing through point.

Question

I have a possible solution to this question. I used the formula found in my textbook to find the equation, but the denominator has a 0. Does this mean that there is no such line?

Answer 1

If a directing vector of the line has coordinates $(2,-1,0)$, the line with this direction passing through $M_0(x_0,y_0,z_0)$ has parametric equation: $$x=x_0+2t,\; y=y_0-t,\; z=z_0$$ which becomes here $$x=1+2t, \; y=-1-t,\; z =3.$$ Now eliminate $t$ between the first two equations: $$t=\frac{x-1}2=-1-y\iff x+2y+1=0,\;z=3.$$

Answer 2

If you have division by zero like this then try clearing denominators first then applying the formula. In this case you'll see that $0(x-1)=0(y+1)=-2(z-3)$ and simplifying we are left with $z-3=0$ so $z=3$.