In the xy-plane, the graph of the given equation x^2 + 58x + y^2 = 0 is a circle. What...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{x^2 + 58x + y^2 = 0}\)
• Need to find: coordinates \(\mathrm{(x, y)}\) of the center

2. INFER the approach needed

• The equation isn't in standard circle form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• To find the center \(\mathrm{(h, k)}\), I need to complete the square
• Focus on the x terms since \(\mathrm{y^2}\) is already in perfect form

3. SIMPLIFY by completing the square

• For \(\mathrm{x^2 + 58x}\), I need to add and subtract \(\mathrm{(58/2)^2 = 29^2 = 841}\)
• Rewrite: \(\mathrm{x^2 + 58x + 841 - 841 + y^2 = 0}\)
• Group: \(\mathrm{(x^2 + 58x + 841) + y^2 = 841}\)
• Factor: \(\mathrm{(x + 29)^2 + y^2 = 841}\)

4. INFER the center coordinates

• Standard form is \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• My equation: \(\mathrm{(x + 29)^2 + (y - 0)^2 = 841}\)
• This means: \(\mathrm{(x - (-29))^2 + (y - 0)^2 = 841}\)
• Therefore: \(\mathrm{h = -29}\) and \(\mathrm{k = 0}\)

Answer: D. \(\mathrm{(-29, 0)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign confusion during completing the square

Students correctly identify they need to complete the square but make a sign error. They might write \(\mathrm{(x - 29)^2}\) instead of \(\mathrm{(x + 29)^2}\), thinking the center x-coordinate is +29 instead of -29. Remember: \(\mathrm{(x + 29)^2 = (x - (-29))^2}\), so the center x-coordinate is -29.

This may lead them to select Choice C. \(\mathrm{(29, 0)}\)

Second Most Common Error:

Conceptual confusion about coordinate identification: Mixing up which number goes with which coordinate

Some students complete the square correctly but then confuse which coordinate is which. They might think the 29 (or -29) goes with the y-coordinate instead of the x-coordinate, since they see the 58 coefficient with x but don't clearly connect it to the final center coordinates.

This may lead them to select Choice A. \(\mathrm{(0, 29)}\) or Choice B. \(\mathrm{(0, -29)}\)

The Bottom Line:

The key challenge is executing completing the square accurately while keeping track of signs, then correctly interpreting what the standard form tells you about the center coordinates.