A circle in the xy-plane has a center at \((3, -7)\). The circle passes through the point \((9, -2)\). The...

1. TRANSLATE the problem information

• Given information:
  • Center of circle: \(\mathrm{(3, -7)}\)
  • Circle passes through point: \(\mathrm{(9, -2)}\)
  • Need to find f in: \(\mathrm{x^2 + y^2 + dx + ey + f = 0}\)

2. INFER the solution approach

• To find f in the general form, we need the complete equation of the circle
• First step: find the radius using the distance from center to the given point
• Second step: write the standard form, then expand to general form

3. SIMPLIFY to find the radius

• Use the distance formula from center \(\mathrm{(3, -7)}\) to point \(\mathrm{(9, -2)}\):
  \(\mathrm{r^2 = (9 - 3)^2 + (-2 - (-7))^2}\)
  \(\mathrm{r^2 = (6)^2 + (5)^2}\)
  \(\mathrm{r^2 = 36 + 25 = 61}\)

4. TRANSLATE to standard form

• The standard form is: \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• Substituting our values: \(\mathrm{(x - 3)^2 + (y + 7)^2 = 61}\)

5. SIMPLIFY by expanding to general form

• Expand each binomial:
  \(\mathrm{(x - 3)^2 = x^2 - 6x + 9}\)
  \(\mathrm{(y + 7)^2 = y^2 + 14y + 49}\)

• Substitute back:
  \(\mathrm{x^2 - 6x + 9 + y^2 + 14y + 49 = 61}\)
  \(\mathrm{x^2 + y^2 - 6x + 14y + 58 = 61}\)
  \(\mathrm{x^2 + y^2 - 6x + 14y - 3 = 0}\)

6. TRANSLATE to identify f

• Comparing with \(\mathrm{x^2 + y^2 + dx + ey + f = 0}\):
  \(\mathrm{f = -3}\)

Answer: C) -3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when expanding \(\mathrm{(y + 7)^2}\) or when moving terms to one side

Students often write \(\mathrm{(y + 7)^2 = y^2 + 14y + 49}\) correctly, but then make mistakes when collecting terms. They might write:
\(\mathrm{x^2 + y^2 - 6x + 14y + 58 = 61}\)
\(\mathrm{x^2 + y^2 - 6x + 14y + 58 - 61 = 0}\)
\(\mathrm{x^2 + y^2 - 6x + 14y - 3 = 0}\) ✓ (correct)

But sometimes calculate 58 - 61 = 3 instead of -3, leading to f = 3.

This may lead them to select Choice D (3).

Second Most Common Error:

Poor INFER reasoning: Not recognizing they need to find the radius first

Some students try to work backwards from the general form or get confused about what information they actually have. They might attempt to plug the point \(\mathrm{(9, -2)}\) directly into \(\mathrm{x^2 + y^2 + dx + ey + f = 0}\) without understanding that they need the radius to properly set up the equation.

This leads to confusion and guessing.

The Bottom Line:

This problem requires systematic progression: find radius → write standard form → expand to general form. Students who skip steps or make careless algebraic errors will select incorrect answers, with sign errors being particularly common.