It is known from algebra that the simultaneous solution set of two equations of the second degree consists of four points. Therefore, two conics will always intersect at four points. These points may all be real and distinct, two real and two imaginary or all imaginary. Two or more points may also coincide.
Example: Find the points of the intersection of the conics and .
We have two given conics
Now to find the point of intersection of these two conics, solve equations (i) and (ii) by using the method of simultaneous equations.
Multiplying equation (ii) by , we get
Now adding equation (i) and (iii), we get
Putting the value of in equation (ii) to get the variable , we have
Thus, the points of intersection of the conics are .