https://file.aiquickdraw.com/mathbot/file/17232551116833m7fb76l.jpg

Thank you for the clarification. The equation of the line is \( 2y = c \), not \( 3y = c \). Let's solve the problem with this updated information.

**Problem:**
"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 the equation \( y = -2x^2 + 9x \), what is the value of \( c \)?"

**Solution:**
1. The line equation is \( 2y = c \). Express \( y \) in terms of \( c \):
   \[
   y = \frac{c}{2}
   \]

2. The parabola equation is given as:
   \[
   y = -2x^2 + 9x
   \]

3. For the line to intersect the parabola at exactly one point, substitute \( y = \frac{c}{2} \) into the equation of the parabola:
   \[
   \frac{c}{2} = -2x^2 + 9x
   \]

4. Rearrange the equation to set it equal to zero:
   \[
   -2x^2 + 9x - \frac{c}{2} = 0
   \]

5. To ensure that the quadratic equation has exactly one solution, the discriminant must be zero. The discriminant \( \Delta \) is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   Here, \( a = -2 \), \( b = 9 \), and the constant term is \( -\frac{c}{2} \).

6. Plug these values into the discriminant formula:
   \[
   \Delta = 9^2 - 4(-2)\left(-\frac{c}{2}\right)
   \]
   \[
   \Delta = 81 - 8\left(\frac{c}{2}\right)
   \]

7. Set the discriminant equal to zero for exactly one solution:
   \[
   81 - 4c = 0
   \]

8. Solve for \( c \):
   \[
   4c = 81
   \]
   \[
   c = \frac{81}{4}
   \]
   \[
   c = 20.25
   \]

The correct value of \( c \) is indeed \( \boxed{20.25} \), which matches the answer preview provided.

Would you like more details, or do you have any questions?

---

**Related Questions:**
1. How does the line equation \( y = \frac{c}{2} \) affect the possible intersection points with a parabola?
2. What role does the discriminant play in determining the number of intersection points?
3. How does changing the coefficient of \( y \) in the line equation affect the solution for \( c \)?
4. Can you determine the vertex of the parabola \( y = -2x^2 + 9x \)?
5.

Learn how to find the value of c in the equation 2y = c such that it intersects the parabola y = -2x^2 + 9x at exactly one point. Follow a step-by-step solution covering equations of lines, quadratic equations, and discriminants. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation