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 in four points. These points may be all real and distinct, two real and two imaginary or all imaginary. Two or more points may also coincide.
Example: Find the points of intersection of the conics and .
We have two given conics
Now to find the point intersection of these two conics, solve the 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 cones are .