Circle M shown is defined by the equation \((x + 4)^2 + (y - 1)^2 = 12\). Circle N (not...

1. TRANSLATE Circle M's equation into useful information

Given equation: \((x + 4)^2 + (y - 1)^2 = 12\)

Comparing to standard form \((x - h)^2 + (y - k)^2 = r^2\):

• Center of Circle M: \((-4, 1)\)
• Remember: \((x + 4) = (x - (-4))\), so \(h = -4\)
• Radius squared: \(r^2 = 12\)
• Radius: \(r = \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\)

2. INFER what we need to find Circle N's equation

The problem asks for Circle N in the form \((x - p)^2 + (y - q)^2 = r\). To write this equation, we need:

• Circle N's center coordinates (which give us p and q)
• Circle N's radius squared (which gives us r)

3. TRANSLATE Circle N's radius from the problem statement

The problem states "Circle N has half the radius of Circle M."

• Circle M's radius = \(2\sqrt{3}\)
• Circle N's radius = \(\frac{1}{2}(2\sqrt{3}) = \sqrt{3}\)

Critical point: "Half the radius" means we halve the radius itself, not the radius squared!

4. SIMPLIFY to find Circle N's radius squared

Since Circle N's radius = \(\sqrt{3}\):

• Radius squared: \(r = (\sqrt{3})^2 = 3\)

This is the value of r in the equation \((x - p)^2 + (y - q)^2 = r\).

5. TRANSLATE Circle N's center from the problem statement

The problem tells us two things about Circle N's center:

1. "Same y-coordinate as Circle M" → \(y = 1\)
2. "Lies on vertical line \(x = 10\)" → \(x = 10\)

• Circle N's center: \((10, 1)\)

6. INFER the values of p and q

Writing Circle N's equation with center \((10, 1)\) and \(r = 3\):

\((x - 10)^2 + (y - 1)^2 = 3\)

Comparing to \((x - p)^2 + (y - q)^2 = r\):

• \(p = 10\)
• \(q = 1\)
• \(r = 3\)

7. SIMPLIFY the final calculation

\(p + q - r = 10 + 1 - 3 = 8\)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Process Skill Gap (INFER): Confusing "half the radius" with "half the radius squared"

Students may see that Circle M has \(r^2 = 12\) and incorrectly think:

• "Half the radius" means \(r^2 = \frac{12}{2} = 6\) for Circle N

This skips the crucial step of finding the actual radius first \((\sqrt{12} = 2\sqrt{3})\) before halving it. Using \(r = 6\) instead of \(r = 3\) in the final calculation:

• \(p + q - r = 10 + 1 - 6 = 5\)

This leads to an incorrect answer of 5 instead of 8.

Second Most Common Error:

Process Skill Gap (TRANSLATE): Misinterpreting which coordinate the vertical line determines

Students may confuse vertical and horizontal lines:

• Thinking "vertical line \(x = 10\)" determines the y-coordinate instead of the x-coordinate
• Using center \((1, 10)\) or \((1, 1)\) instead of \((10, 1)\)

If they use center \((1, 1)\), they get \(p = 1\), leading to:

• \(p + q - r = 1 + 1 - 3 = -1\)

This confusion causes them to arrive at an incorrect negative answer.

The Bottom Line:

This problem requires careful attention to the distinction between radius and radius squared throughout the solution. The phrase "half the radius" must be interpreted at the radius level \((2\sqrt{3} \div 2 = \sqrt{3})\), not at the radius-squared level \((12 \div 2 = 6)\). Additionally, correctly identifying the center coordinates from verbal descriptions requires understanding geometric terms like "vertical line."