Point P lies on a circle centered at the origin with radius 7sqrt(10). The x-coordinate of P is 7. What...

1. INFER the mathematical relationship

• Given information:
  • Circle centered at origin with radius \(7\sqrt{10}\)
  • Point P(7, y) lies on this circle
• Key insight: Any point on a circle centered at the origin satisfies \(\mathrm{x}^2 + \mathrm{y}^2 = \mathrm{r}^2\)

2. TRANSLATE into the circle equation

• Set up the equation: \(\mathrm{x}^2 + \mathrm{y}^2 = (7\sqrt{10})^2\)
• We need to find the absolute value of y when x = 7

3. SIMPLIFY the radius calculation

• Calculate \((7\sqrt{10})^2\):
  • \((7\sqrt{10})^2 = 7^2 \times (\sqrt{10})^2\)
  • \(= 49 \times 10 = 490\)
• So our equation becomes: \(\mathrm{x}^2 + \mathrm{y}^2 = 490\)

4. SIMPLIFY by substitution and solving

• Substitute x = 7: \(7^2 + \mathrm{y}^2 = 490\)
• This gives us: \(49 + \mathrm{y}^2 = 490\)
• Subtract 49 from both sides: \(\mathrm{y}^2 = 441\)
• Take the square root: \(|\mathrm{y}| = \sqrt{441} = 21\)

Answer: D (21)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making computational errors with \((7\sqrt{10})^2\)

Students often struggle with compound expressions like \((7\sqrt{10})^2\). They might compute it incorrectly as:

• \(7^2 + (\sqrt{10})^2 = 49 + 10 = 59\), or
• \(7 \times \sqrt{10} \approx 22.1\), then \((22.1)^2 \approx 488\)

These calculation errors cascade through the rest of the problem, leading to incorrect values for \(\mathrm{y}^2\) and ultimately wrong final answers. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Weak INFER reasoning: Not recognizing the need to apply the circle equation

Some students see the coordinate geometry setup but don't immediately connect it to the fundamental circle equation \(\mathrm{x}^2 + \mathrm{y}^2 = \mathrm{r}^2\). Instead, they might try to use distance formulas inappropriately or attempt trigonometric approaches. Without the correct starting equation, they get stuck early in the problem and resort to guessing.

The Bottom Line:

This problem tests whether students can bridge coordinate geometry concepts with algebraic manipulation. The key is recognizing the circle equation as the starting point, then executing the algebra carefully through multiple calculation steps.