In the xy-plane, a circle has equation x^2 + y^2 - 14x + 10y = k, where k is a...

1. TRANSLATE the given information

• Given information:
  • Circle equation: \(\mathrm{x^2 + y^2 - 14x + 10y = k}\)
  • Point (3, -5) lies on the circle
• What this tells us: Since the point lies on the circle, its coordinates must satisfy the equation

2. INFER the solution strategy

• We need the radius, which requires the standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• First find k using the given point, then complete the square

3. TRANSLATE and substitute to find k

• Substitute \(\mathrm{x = 3, y = -5}\) into the equation:
  \(\mathrm{3^2 + (-5)^2 - 14(3) + 10(-5) = k}\)
• Calculate:
  \(\mathrm{9 + 25 - 42 - 50 = k}\)
• Therefore:
  \(\mathrm{k = -58}\)

4. SIMPLIFY by completing the square

• Start with: \(\mathrm{x^2 + y^2 - 14x + 10y = -58}\)
• Group terms: \(\mathrm{(x^2 - 14x) + (y^2 + 10y) = -58}\)
• Complete the square for x: \(\mathrm{x^2 - 14x = (x - 7)^2 - 49}\)
• Complete the square for y: \(\mathrm{y^2 + 10y = (y + 5)^2 - 25}\)
• Substitute back: \(\mathrm{(x - 7)^2 - 49 + (y + 5)^2 - 25 = -58}\)

5. SIMPLIFY to standard form

• Combine constants:
  \(\mathrm{(x - 7)^2 + (y + 5)^2 = -58 + 49 + 25}\)
• Final form:
  \(\mathrm{(x - 7)^2 + (y + 5)^2 = 16}\)

6. INFER the radius value

• Since the standard form is \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\), we have \(\mathrm{r^2 = 16}\)
• Therefore \(\mathrm{r = 4}\)

Answer: B) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize they need to complete the square but forget that the final result gives \(\mathrm{r^2}\), not r directly.

After correctly completing the square to get \(\mathrm{(x - 7)^2 + (y + 5)^2 = 16}\), they see "16" and immediately think the radius is 16. They fail to recognize that this coefficient represents \(\mathrm{r^2}\), requiring them to take the square root for the actual radius.

This may lead them to select Choice E (16).

Second Most Common Error:

Poor TRANSLATE reasoning: Students substitute the point incorrectly or make arithmetic errors when calculating k.

For example, they might write \(\mathrm{(-5)^2}\) as -25 instead of +25, or make sign errors when substituting into \(\mathrm{-14x + 10y}\). This leads to an incorrect value of k, which propagates through the entire solution and results in a wrong radius calculation.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem requires careful attention to the distinction between \(\mathrm{r^2}\) and r. Many students complete the algebraic work correctly but miss the final conceptual step of taking the square root.