Locus of a variable line intersecting 3 non-parallel lines.

Question

Can anybody help me solve this problem? Find the locus of a variable st line intersecting the 3 lines $x=a,y=0$ and $y=a,z=0$ and $z=a,x=0$.What type of surface it is?(How would it look in space?) I had started like this. let the variable line be $\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$.Then I took the 3 points where it cuts the 3 lines as $(a,0,z),(x,a,0)$ and $(0,y,a)$.Then $l,m,n$ could be replaced by d.r. obtained from any of these 2 pts,then I satisfied the variable line with the 3rd one.But it is not easy to get the eqn of the required surface like this.How do I do it.Also I want to know geometrically what is going on and how the surface would look.If anybody has any intuitive idea of the problem then please answer to this question.

Answer 1

The similar question that you found provides a way to classify the surface without finding an equation for the one in yours. In that question, the surface turns out to be a quadric. By construction, we know that it is a ruled surface, so it must be either a hyperboloid of one sheet, a cone (degenerate hyperboloid) or a cylinder. Observe that each of the fixed lines used to construct the surface lie on the surface (verify this!), so it is a doubly-ruled quadric, which is a hyperboloid of one sheet.

Now, for your problem, when $a\ne0$ we also have three skew lines, which can be mapped to the three lines in the other problem via some affine transformation. These transformations preserve incidence relations, so the surface generated by the three lines in your problem is an affine image of the one in the other problem. Affine transformations don’t change the nature of a quadric, therefore your surface is also a hyperboloid of one sheet when $a=0$.

When $a=0$, the construction in the problem breaks down: the three lines intersect at the origin, so given a point on any of the three lines, the only line through that point that intersects all three is that line itself, so the locus is just the three original lines. On the other hand, if we look at what happens to the family of surfaces as $a$ approaches zero, we will find that they approach a cone with vertex at the intersection point.

To find a Cartesian equation of the surface, the simplest way to proceed is described in achille hui’s answer to the other question, which is a matter of computing a certain $3\times3$ determinant.