# Parametars B and D of line

## Homework Statement

For which B and D the line $$\left\{\begin{matrix} x-2y+z-9=0 & \\ 3x+By+z+D=0 & \end{matrix}\right.$$ is in xy-plane ?

## The Attempt at a Solution

I found the line:

$$\frac{x-\frac{-D+9}{2}}{2-B}=\frac{y}{2}=\frac{z-\frac{D+27}{2}}{B+6}$$

What should I do next?

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tiny-tim
Homework Helper
$$\frac{x-\frac{-D+9}{2}}{2-B}=\frac{y}{2}=\frac{z-\frac{D+27}{2}}{B+6}$$
Hi Physicsissuef! Hint: you want z to be constant, but you don't want y to be constant.

So … ? If it is in the xy-plane probably B+6=0.

tiny-tim
Homework Helper
But how do you stop y being infinite, then? A point in xy plane have coordinates (x,y,0). A line in xy plane probably have $a(a_1,a_2,0)$. a is parallel vector to the line, in this case $a(2-B , 2 , B+6)$. And I don't understand you what are you talking to me... HallsofIvy
Homework Helper
I think you are putting far too much emphasis on plugging into formulas rather that thinking about what the problem says. In the xy plane, z is not just constant- it is specifically 0. If z= 0 then your equations become 3x- 2y- 9= 0 and 3x+ By+ D= 0. In the xy plane, those are each the equation of a line. For what B and D are those the same line?

B=-6 and D=-27. But why they need to be same?

HallsofIvy
Homework Helper

Ohh.... I see, thanks. And is it correct that "a" should have coordinates $a_1,a_2,0$?

tiny-tim
Homework Helper
And I don't understand you what are you talking to me... I was referring to the fact that your equation y/2 = (z - (D+27)/2)/(B + 6) seems to make y fixed and infinite if B + 6 = 0, while you want y to be able to take any value. A point in xy plane have coordinates (x,y,0). A line in xy plane probably have $a(a_1,a_2,0)$. a is parallel vector to the line, in this case $a(2-B , 2 , B+6)$.
Well, yes, the line is parallel to a line through the origin with (a_1,a_2,0).

But you're concentrating too much on formulas, as HallsofIvy said.

btw, his method is a lot easier than the one you chose … if you do that in the exam, the examiner wil get the impression you're using the formula without really understanding it.

Hehe... I used the formula because I was desperate, I forgot that z=0. Thanks for the help.