\((\mathrm{x} + 5)^2 + (\mathrm{y} - 4)^2 = 25\) x = -11 If the given equations are graphed in the...

1. TRANSLATE the given equations into their geometric meanings

• Given information:
  • \((\mathrm{x} + 5)^2 + (\mathrm{y} - 4)^2 = 25\) (circle equation in standard form)
  • \(\mathrm{x} = -11\) (vertical line equation)

• What this tells us:
  • Circle has center \((-5, 4)\) and radius \(\sqrt{25} = 5\)
  • Vertical line passes through all points where \(\mathrm{x} = -11\)

2. INFER the most efficient solution approach

• Two viable approaches: graphical analysis or algebraic substitution
• Graphical approach may be faster since we can visualize the relationship
• Let's use graphical reasoning first, then verify algebraically

3. INFER the circle's position on the coordinate plane

• With center at \((-5, 4)\) and radius \(5\):
  • Leftmost point: \(\mathrm{x} = -5 - 5 = -10\)
  • Rightmost point: \(\mathrm{x} = -5 + 5 = 0\)
• The circle occupies x-coordinates from \(-10\) to \(0\)

4. Compare the line's position to the circle

• The vertical line \(\mathrm{x} = -11\) is located at \(\mathrm{x} = -11\)
• Since \(-11 \lt -10\), this line is entirely to the left of the circle
• No intersection occurs

5. Verify algebraically by substituting \(\mathrm{x} = -11\)

SIMPLIFY the substitution:

\((-11 + 5)^2 + (\mathrm{y} - 4)^2 = 25\)

\((-6)^2 + (\mathrm{y} - 4)^2 = 25\)

\(36 + (\mathrm{y} - 4)^2 = 25\)

\((\mathrm{y} - 4)^2 = -11\)

• Since squares of real numbers cannot be negative, no real solutions exist

Answer: (A) Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize \((\mathrm{x} + 5)^2 + (\mathrm{y} - 4)^2 = 25\) as a circle equation or incorrectly identify the center and radius.

Some students might think the center is \((5, -4)\) instead of \((-5, 4)\), leading them to believe the circle extends from \(\mathrm{x} = 0\) to \(\mathrm{x} = 10\). In this incorrect scenario, they might think \(\mathrm{x} = -11\) could intersect the circle, potentially leading them to select Choice (C) (Exactly two) or get confused and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make computational errors during algebraic substitution, particularly when calculating \((-6)^2\) or handling the negative result.

A student might incorrectly compute \((-6)^2\) as \(-36\) instead of \(36\), leading to \((\mathrm{y} - 4)^2 = 25 + 36 = 61\), which would give real solutions. This could lead them to select Choice (C) (Exactly two) since quadratic equations typically yield two solutions.

The Bottom Line:

This problem requires both geometric visualization skills and algebraic manipulation. Success depends on correctly translating the standard circle form and recognizing when no real solutions exist.