If the line, $$\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z+4}{3}$$ lies in the plane, $$lx + my - z = 9$$, then $$l^2 + m^2$$ is equal to

We have the line written in symmetric (or two-point) form

$$\frac{x-3}{2} \;=\; \frac{y+2}{-1} \;=\; \frac{z+4}{3}.$$

To change this into parametric form, we introduce a real parameter, say $$t$$, and set each ratio equal to $$t$$. Thus

$$\frac{x-3}{2}=t,\qquad \frac{y+2}{-1}=t,\qquad \frac{z+4}{3}=t.$$

Solving each relation for the corresponding coordinate one by one, we obtain

$$x-3 = 2t \;\;\Longrightarrow\;\; x = 3 + 2t,$$

$$y + 2 = -t \;\;\Longrightarrow\;\; y = -2 - t,$$

$$z + 4 = 3t \;\;\Longrightarrow\;\; z = -4 + 3t.$$

So every point on the line can be written as

$$\bigl(x,\,y,\,z\bigr) = \bigl(3 + 2t,\; -2 - t,\; -4 + 3t\bigr),\qquad t\in\mathbb R.$$

Now the plane is given by the equation

$$lx + my - z = 9.$$

Because the entire line lies in the plane, every point obtained from the parametric form must satisfy the plane equation, no matter what real value $$t$$ takes. We therefore substitute $$x=3+2t,\; y=-2-t,\; z=-4+3t$$ into the plane:

$$l(3 + 2t) \;+\; m(-2 - t) \;-\; (-4 + 3t) \;=\; 9.$$

Expanding each product carefully, we get

$$3l + 2lt \;-\; 2m - mt \;+\; 4 - 3t \;=\; 9.$$

Next we collect the coefficients of the parameter $$t$$ and the constant terms separately:

$$\bigl(2l - m - 3\bigr)t \;+\; \bigl(3l - 2m + 4\bigr) \;=\; 9.$$

This identity has to hold for every value of $$t$$. The only way a linear expression in $$t$$ can equal a constant (independent of $$t$$) is if the coefficient of $$t$$ itself is zero and the constant term equals the given constant on the right. Hence we must have the simultaneous conditions

$$2l - m - 3 = 0,$$

$$3l - 2m + 4 = 9.$$

We now solve this pair of linear equations in $$l$$ and $$m$$.

From the first equation we isolate $$m$$:

$$2l - m - 3 = 0 \;\;\Longrightarrow\;\; m = 2l - 3.$$

Substituting this value of $$m$$ into the second equation, we have

$$3l \;-\; 2(2l - 3) \;+\; 4 \;=\; 9.$$

Simplifying step by step, we get

$$3l - 4l + 6 + 4 = 9,$$

$$-l + 10 = 9,$$

$$-l = -1,$$

$$l = 1.$$

Returning to $$m = 2l - 3$$ and substituting $$l = 1$$, we find

$$m = 2(1) - 3 = -1.$$

Finally we evaluate $$l^2 + m^2$$:

$$l^2 + m^2 = 1^2 + (-1)^2 = 1 + 1 = 2.$$

Hence, the correct answer is Option B.