Points of Intersection Between a Parabola and a Conic

1. Use the appropriate method (comparison, substitution, or elimination) to obtain an equation with one variable.

2. Manipulate the equation so that it equals |0.|

3. Solve the equation to find the value(s) of the isolated variable.

4. Replace the value(s) obtained in one of the original equations to obtain the value(s) of the other variable.

5. Write the coordinates of the point(s) of intersection.

Unlike the intersection between a line and a conic, there are five possible cases regarding the number of solutions:

• the parabola and the conic do not intersect;

• the parabola and the conic intersect only at one place, called the point of tangency; or

• the parabola and the conic intersect in two, three, or four distinct places.

In the following interactive video, select a conic section and drag the cursor to study possible cases.

Determine the coordinates of the point(s) of intersection between the parabola |y^2=-16x| and the circle |x^2+y^2=36.|

Determine the coordinates of the point(s) of intersection between the parabola |x^2=12(y+3)| and the ellipse |\dfrac{x^2}{25}+\dfrac{y^2}{9}=1.|

Determine the coordinates of the point(s) of intersection between the parabola |x^2=-4(y-2{.}75)| and hyperbola |\dfrac{x^2}{4}-\dfrac{y^2}{9}=1.|

Determine the coordinates of the point(s) of intersection between the parabolas |(y+4)^2=8(x-2)| and |(y-1)^2=-16(x-10).|