In the xy-plane, the graph of the parabola with equation y = x^2 + 25 does not intersect the graph...

1. TRANSLATE the intersection condition

• Given information:
  • Parabola: \(\mathrm{y = x^2 + 25}\)
  • Line: \(\mathrm{y = kx}\)
  • These graphs do NOT intersect
  • k is an integer
• What "no intersection" means: The system of equations has no real solutions

2. INFER the mathematical approach

• To find intersections, we set the equations equal
• For no intersections, the resulting equation must have no real solutions
• This connects to discriminant analysis of quadratic equations

3. Set up the equation system

Set the parabola and line equal:

\(\mathrm{x^2 + 25 = kx}\)

4. SIMPLIFY to standard quadratic form

\(\mathrm{x^2 + 25 = kx}\)
\(\mathrm{x^2 - kx + 25 = 0}\)

5. APPLY discriminant condition for no real solutions

For \(\mathrm{ax^2 + bx + c = 0}\), no real solutions when \(\mathrm{discriminant \lt 0}\)
Here: \(\mathrm{a = 1, b = -k, c = 25}\)

\(\mathrm{Discriminant = b^2 - 4ac \lt 0}\)
\(\mathrm{(-k)^2 - 4(1)(25) \lt 0}\)
\(\mathrm{k^2 - 100 \lt 0}\)

6. SIMPLIFY the inequality

\(\mathrm{k^2 \lt 100}\)
Taking square roots: \(\mathrm{-10 \lt k \lt 10}\)

7. APPLY CONSTRAINTS for integer solutions

Since k must be an integer: \(\mathrm{k \in \{-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}}\)

The greatest possible value is \(\mathrm{k = 9}\).

Answer: B) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misunderstand what "no intersection" means mathematically. They might try to solve for intersection points instead of setting up the no-solution condition.

This leads to confusion because they're looking for specific x-values rather than the condition that prevents any x-values from existing. They may randomly guess among the answer choices.

Second Most Common Error:

Missing conceptual knowledge about discriminants: Students may correctly set up \(\mathrm{x^2 - kx + 25 = 0}\) but not remember that \(\mathrm{discriminant \lt 0}\) means no real solutions.

Without this connection, they cannot proceed systematically and may attempt to factor or use other inappropriate methods. This may lead them to select Choice C (10) by incorrectly thinking k can equal 10.

The Bottom Line:

This problem requires students to translate a geometric condition (no intersection) into an algebraic condition (no real solutions), then apply discriminant analysis. The key insight is recognizing that "no intersection" means the discriminant must be negative.