What is the geometrical meaning of equating two curves, in general?
For e.g equating a circle with another gives a line passing through their common points of intersection. Equating two planes gives their line of intersection.
I understand this can be proved but I don't understand it intuitively. E.g what would I get on equating a circle with a line, or a circle with a parabola etc.
On
What you call "equating two curves" is actually a statement about the locus of all points $(x,y)$ that satisfy an equation in two variables. This is a special case of what is called a "pencil of curves". For example, suppose $f(x,y)=0,\;g(x,y)=0$ are the equations of two circles in the plane. If $a,b$ are any two reals, not both $0$, then equating the linear combination $\;af(x,y)+bg(x,y)\;$ to $\;0\;$ is the equation of a pencil of circles that pass through the points of intersection of the two original circles. The special case $\;a=1,b=-1\;$ corresponds to your question about two circles. Notice that if $\;f(x,y),\;g(x,y)\;$ are quadratic polynomials, then any linear combination is also quadratic at most.
The “equation of curve” is not directly a statement about the curve. It is actually an equation that a point on the curve needs to satisfy. For example, $f(x,y)=x^2+y^2-25=0$ is a constraint that points $(x,y)$ need to satisfy. $(3,4)$ meets that criterion, whereas $(2,10)$ does not. The set of all points that satisfies this condition is the circle $C$.
Similarly $g(x,y)=2x+3y-1=0$ specifies the set of points that form that line $L$. Now, if a point $p=(a,b)$ is in the intersection of both curves, then it lies in both $C$ and $L$. So it must satisfy both sets of constraints. $f(a,b)=0$ and $g(a,b)=0$. So you can set $f(a,b)=g(a,b)=0$ and solve it to find all conditions that $p=(a,b)$ must meet. There might be more than one such $p$. The solution gives you the whole set of such $p$ points.
Please note that to find the intersection, you must use the final equality (equating to zero) $$f(a,b)=g(a,b)\mathbf{=0}$$
Else, you will get a locus.
For example, if f=0 and g=0 represent the unit circles centered at p=(-1,0) and q=(1,0) respectively and radius of 13 each. Here $f(x,y)=(x+1)^2+y^2-13^2$ and $g(x,y)=(x-1)^2+y^2-13^2$, then think about what $f=g$ means.
Here $f$ and $g$ represent the (square of) distance of a point from the respective centers (offset by a constant). That constant is the square of the radius.
Then if you solve $f=g=0$, then we are asking “what are all the points at a distance of 13 from both p and q?” The answer is just two points $(0,\pm 12)$.
But if you just solve $f=g$, we are only asking “what is the set of points equidistant from p and q” and the answer to that of course, is a whole line (the y axis).
Now in this case, $f$ and $g$ had meanings in terms of distances. That may not be the case for all equations of curves. So basically, whenever you equate $f$ and $g$, you need to ask yourselves what $f$ and $g$ mean in the first place and you will find that the your solution is simply turn set of points that satisfies both meanings.