Points of Intersection Between a Line and a Conic

1. Use the substitution method to obtain an equation with one variable.

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

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

4. Replace the value(s) obtained into 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 parabola and a conic, there are three possible cases regarding the number of solutions:

• the line and the conic do not intersect;

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

• the line and the conic intersect at two distinct places.

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

Determine the coordinates of the point(s) of intersection between the line |y=2x+5| and the circle |x^2+y^2=10.|

Determine the coordinates of the point(s) of intersection between the line |y =-2x+6| and the ellipse |\dfrac{x^2}{36}+\dfrac{y^2}{49}=1.|

Determine the coordinates of the point(s) of intersection between the line |y=2x-13| and the hyperbola |\dfrac{x^2}{25}-\dfrac{y^2}{100}=1.|

Determine the coordinates of the point(s) of intersection between the line |y=4x-7| and the parabola |(x-4)^2=3(y+6).|