Question

In the xy-plane, a line with equation 2y=c for some constant c intersects a parabola at exactly one point. If the parabola has equation y=-2x²+9x, what is the value of c ?

Answer 1

To find the value of c, we need to determine the point at which the line 2y = c intersects the parabola y = -2x² + 9x.

Step-by-step explanation:

To find the value of c, we need to determine the point at which the line 2y = c intersects the parabola y = -2x² + 9x. Substitute the equation of the line into the equation of the parabola:

2y = c

-2x² + 9x = c

Rearrange the equation to get:

2x² - 9x + c = 0

Since the line intersects the parabola at exactly one point, the quadratic equation should have exactly one solution. For a quadratic equation to have one solution, the discriminant (b² - 4ac) must equal zero. Plugging in the values:

(-9)² - 4(2)(c) = 0

Solving this equation will give us the value of c.