In the xy-plane, a line k intersects the graph of y = sqrt(100 - x^2) at x = -8 and...

1. TRANSLATE the problem information

• Given information:
  • Line k intersects \(\mathrm{y = \sqrt{100 - x^2}}\) at \(\mathrm{x = -8}\) and \(\mathrm{x = 6}\)
  • Need to find the slope of line k
• This tells us we have two x-coordinates where the line crosses the curve

2. INFER the approach

• To find a line's slope, we need two points on that line
• Since we know the x-coordinates of intersection, we can find the corresponding y-coordinates using the given function
• Then we can apply the slope formula

3. TRANSLATE the x-values into coordinate pairs

• At \(\mathrm{x = -8}\): \(\mathrm{y = \sqrt{100 - (-8)^2}}\)
  \(\mathrm{= \sqrt{100 - 64}}\)
  \(\mathrm{= \sqrt{36}}\)
  \(\mathrm{= 6}\)
  First point: \(\mathrm{(-8, 6)}\)
• At \(\mathrm{x = 6}\): \(\mathrm{y = \sqrt{100 - 6^2}}\)
  \(\mathrm{= \sqrt{100 - 36}}\)
  \(\mathrm{= \sqrt{64}}\)
  \(\mathrm{= 8}\)
  Second point: \(\mathrm{(6, 8)}\)

4. SIMPLIFY using the slope formula

• Apply \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{8 - 6}{6 - (-8)}}\)
  \(\mathrm{= \frac{2}{14}}\)
  \(\mathrm{= \frac{1}{7}}\)

Answer: B (1/7)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when evaluating the square roots or when calculating the final slope.

For example, they might incorrectly compute \(\mathrm{\sqrt{36} = 8}\) instead of 6, or \(\mathrm{\sqrt{64} = 6}\) instead of 8. This leads to wrong coordinates and consequently an incorrect slope calculation. Another common mistake is computing the slope formula incorrectly, such as getting the denominator wrong: \(\mathrm{(6 - (-8)) = 14}\), not 2.

This may lead them to select Choice A (-1/7) or Choice D (7).

The Bottom Line:

This problem tests whether students can systematically work through function evaluation and slope calculation without making computational errors. The function itself isn't complex, but the multi-step arithmetic creates opportunities for mistakes.