Given a set of two equations (one linear and one quadratic in $x$ and $y$) as follows:-
$$ax + by + c = 0 \tag 1$$
$$Ax^2 + Bxy +Cy^2 + Dx + Ey + F = 0 \tag 2$$
What are the conditions that can be imposed on the coefficients such that the solution of x (or y) has exactly one root (not equal or double repeated roots).
The following is an example to illustrate my point:-
$$x + y = 6 \tag 3$$
$$x^2 - y^2 = 12 \tag 4$$
At one glance of the above, we are expecting two roots (because one equation is quadratic) but it turns out that we can get $(x, y) = (4, 2)$ only.
$a x + by + c = 0$ is a line.
$A x^2 + B xy + C y^2 + Dx + Ey +F = 0$ Is some sort of conic section, that has been rotated off of the standard.
If $B^2 - 4AC = 0$ you have a parabola.
I line will intersect a parabola it two places, unless the line is tangent to the parabola (one place), is parallel to the axis of symmetry (one place), or fails to intersect (no solution).
If $B^2 - 4AC < 0$ you have an ellipse.
A line will intersect an ellipse in two places, unless it is tangent, or it fails to intersect.
If $B^2 - 4AC >0$ you have an hyperbola.
A line will intersect a hyperbola in two places, unless the line is parallel to an asymptote, or is an asymptote.
In the example, the quadratic is a hyperbola, and the line is parallel to its asymptote.