In the xy -plane, a line with equation 2 y = c for some constant c intersects a parabola at

Question

In the xy -plane, a line with equation 2 y = c for some constant c intersects a parabola at exactly one point. If the parabola has equation y = - 2 x 2 + 9 x , what is the value of c ?

Accepted Answer

The correct answer is 81 4 . The given linear equation is 2 y = c . Dividing both sides of this equation by 2 yields y = c 2 . Substituting c 2 for y in the equation of the parabola yields c 2 = - 2 x 2 + 9 x . Adding 2 x 2 and - 9 x to both sides of this equation yields 2 x 2 - 9 x + c 2 = 0 . Since it&rsquo;s given that the line and the parabola intersect at exactly one point, the equation 2 x 2 - 9 x + c 2 = 0 must have exactly one solution. An equation of the form A x 2 + B x + C = 0 , where A , B , and C are constants, has exactly one solution when the discriminant, B 2 - 4 A C , is equal to 0 . In the equation 2 x 2 - 9 x + c 2 = 0 , where A = 2 , B = - 9 , and C = c 2 , the discriminant is - 9 2 - 4 2 c 2 . Setting the discriminant equal to 0 yields - 9 2 - 4 2 c 2 = 0 , or 81 - 4 c = 0 . Adding 4 c to both sides of this equation yields 81 = 4 c . Dividing both sides of this equation by 4 yields c = 81 4 . Note that 81/4 and 20.25 are examples of ways to enter a correct answer.

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