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?

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.

Learn how to determine the value of c where a line intersects a parabola at exactly one point. This problem involves solving quadratic equations and applying the discriminant condition. Suitable for high school students in Grades 10-12.

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