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We are given two equations in the problem:

1. The line: \( 2y = c \), where \( c \) is a constant.
2. The parabola: \( y = -2x^2 + 9x \).

The goal is to find the value of \( c \) such that the line and the parabola intersect at exactly one point.

### Step 1: Express the line equation in terms of \( y \)

From the equation \( 2y = c \), solve for \( y \):

\[
y = \frac{c}{2}
\]

### Step 2: Set the equations equal to each other

Since both equations describe \( y \), set the right-hand side of the parabola equal to the right-hand side of the line equation:

\[
-2x^2 + 9x = \frac{c}{2}
\]

Multiply both sides of the equation by 2 to eliminate the fraction:

\[
-4x^2 + 18x = c
\]

### Step 3: Rearrange the equation

Rearrange the equation into a standard quadratic form:

\[
-4x^2 + 18x - c = 0
\]

### Step 4: Use the condition for exactly one intersection

For the line to intersect the parabola at exactly one point, the quadratic equation must have exactly one solution. This happens when the discriminant is zero. The discriminant \( \Delta \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[
\Delta = b^2 - 4ac
\]

Here, the quadratic equation is:

\[
-4x^2 + 18x - c = 0
\]

Thus, \( a = -4 \), \( b = 18 \), and \( c = -c \). The discriminant is:

\[
\Delta = 18^2 - 4(-4)(-c)
\]

Simplifying:

\[
\Delta = 324 - 16c
\]

Set the discriminant equal to zero for exactly one solution:

\[
324 - 16c = 0
\]

### Step 5: Solve for \( c \)

\[
16c = 324
\]

\[
c = \frac{324}{16}
\]

\[
c = 20.25
\]

Thus, the value of \( c \) is \( 20.25 \).

Would you like further explanation on any step? Here are some related questions for further exploration:

1. How does the discriminant determine the number of solutions to a quadratic equation?
2. What happens if the discriminant is positive or negative?
3. How do we handle cases where the intersection results in two points instead of one?
4. Can this method be extended to other types of curves, such as ellipses or hyperbolas?
5. How would the result change if the line equation was different, such as \( y = c \)?

**Tip**: When solving systems of equations with parabolas, always consider the discriminant for determining the nature of intersections!

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 = -2x^2 + 9x, what is the value of c?

Discover how to find the constant c such that a line with equation 2y = c intersects a parabola at exactly one point. This step-by-step solution uses the quadratic discriminant to solve the system of equations. Ideal for Grades 10-12.

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