Find the Points of Intersection of a Parabola with a Line




This is tutorial on finding the points of intersection of a parabola with a line; general solution.

Example 1: Find the points of intersection of the parabola with the line given respectively by their equations

y = 2 x 2 + 4 x - 3

2y + x = 4

Solution to Example 1:

  • We first solve the linear equation for y as follows:
    y = - (1 / 2) x + 2

  • We now substitute y in the equation of the parabola by - (1 / 2) x + 2 as follows

    - (1 / 2) x + 2 = 2 x 2 + 4 x - 3

  • We now group like terms

    2 x2 + (9 / 2) x - 5 = 0

  • Solve the above quadratic equation for x to obtain two solutions

    x = (- 9 - √(241)) / 8 and x = (- 9 + √(241)) / 8

  • We now substitute the values of x obtained above into the equation y = - (1 / 2) x + 2 to obtain the values for y as follows

    y = (41 + √(241)) / 16

    and y = (41 - √(241)) / 16

  • The two points of intersection of the two circless are given by

    ((- 9 - √(241)) / 8 , (41 + √(241)) / 16 ) and ((- 9 + √(241)) / 8 , (41 - √(241)) / 16 )

    Approximated as:(-3.06 , 3.53 ) and (0.82 , 1.59)

Shown below is the graph of the parabola, the line and the two points of intersection.

Points of intersection of a parabola and a line



More links and references related to the above topics.

Share
Popular Pages