To solve for the value of $c$ where the line $2y = c$ intersects the parabola $y = -2x^2 + 9x$ at exactly one point, we'll follow these steps:

Step 1: Express the equation of the line in terms of $y$

Given the equation of the line: $2y = c$ Divide both sides by 2 to solve for $y$: $y = \frac{c}{2}$

Step 2: Set up the equation for the intersection points

For the line to intersect the parabola, the $y$ values from both equations must be equal. Thus, we equate the parabola equation to the line equation: $-2x^2 + 9x = \frac{c}{2}$ Multiply the entire equation by 2 to eliminate the fraction: $-4x^2 + 18x = c$ Rearrange to form a standard quadratic equation: $-4x^2 + 18x - c = 0$

Step 3: Determine the condition for exactly one intersection

A quadratic equation $ax^2 + bx + c = 0$ has exactly one solution if its discriminant is zero. The discriminant $\Delta$ is given by: $\Delta = b^2 - 4ac$ For the quadratic equation $-4x^2 + 18x - c = 0$, we have:

• $a = -4$
• $b = 18$
• $c = -c$ (as a constant)

The discriminant is: $\Delta = 18^2 - 4(-4)(-c) = 324 - 16c$ Set the discriminant equal to zero for exactly one intersection: $324 - 16c = 0$

Step 4: Solve for $c$

$16c = 324$ $c = \frac{324}{16} = 20.25$

Thus, the value of $c$ is $\boxed{20.25}$.

Would you like more details or have any questions about this?

Here are some related questions you might find interesting:

1. How would the solution change if the parabola's equation had a different coefficient for $x^2$?
2. What is the significance of the discriminant in determining the number of intersections?
3. How would the intersection points change if $c$ were greater or less than 20.25?
4. Can you derive a general formula for $c$ in terms of the coefficients of the quadratic equation?
5. What would happen if the parabola opened upwards instead of downwards?

Tip: Always check the discriminant when dealing with the intersection of a line and a parabola to determine the number of intersection points.