Equation of a curve passing through the point of intersection of two curves.

Question

It is given in my book that, equation of any curve passing through the point of intersection of two curves $C_1=0$ and $C_2=0$ , is given by $a \cdot C_1+b \cdot C_2=0$, where $a$ and $b$ are real parameters. I wanted to know the proof of this fact. I have proved the above result specifically for lines and circles but I am not able to prove the result in general. Please guide me.

Answer 1

Your curves are characterised by $C_1(x,y) = 0$ and $C_2(x,y) = 0$. If some point $x, y$ satisfies these, then $aC_1(x,y) + bC_2(x,y) = a \cdot 0 + b \cdot 0 = 0$, as required.