A parabola in the xy-plane has its focus at \((2, -3)\) and its directrix is the line y = 1....

1. TRANSLATE the problem information

• Given information:
  • Focus: \((2, -3)\)
  • Directrix: \(\mathrm{y = 1}\)
  • Need: Equation of parabola

• What this tells us: We need to use the definition that any point on a parabola is equidistant from the focus and directrix.

2. TRANSLATE the definition into mathematics

• For any point \((x, y)\) on the parabola:
  • Distance to focus \((2, -3)\): \(\sqrt{(x-2)^2 + (y+3)^2}\)
  • Distance to directrix \(\mathrm{y = 1}\): \(|y - 1|\)
• Setting equal: \(\sqrt{(x-2)^2 + (y+3)^2} = |y - 1|\)

3. INFER the parabola orientation

• Since focus \((2, -3)\) is at \(\mathrm{y = -3}\) and directrix is at \(\mathrm{y = 1}\), the focus lies below the directrix
• This means the parabola opens downward
• Points on the parabola satisfy \(\mathrm{y \lt 1}\), so \(|y - 1| = 1 - y\)

4. SIMPLIFY by squaring both sides

\(\sqrt{(x-2)^2 + (y+3)^2} = 1 - y\)

• Squaring: \((x-2)^2 + (y+3)^2 = (1-y)^2\)

5. SIMPLIFY the algebraic expansion

• Left side: \((x-2)^2 + y^2 + 6y + 9\)
• Right side: \((1-y)^2 = 1 - 2y + y^2\)
• Setting equal: \((x-2)^2 + y^2 + 6y + 9 = 1 - 2y + y^2\)

6. SIMPLIFY to isolate the standard form

• Cancel \(y^2\) terms: \((x-2)^2 + 6y + 9 = 1 - 2y\)
• Combine y terms: \((x-2)^2 = 1 - 2y - 6y - 9\)
• Simplify: \((x-2)^2 = -8y - 8 = -8(y + 1)\)

Answer: A. \((x - 2)^2 = -8(y + 1)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing which direction the parabola opens, leading to incorrect handling of the absolute value \(|y - 1|\).

Students might think \(|y - 1|\) always equals \(y - 1\), regardless of the parabola's orientation. This leads to the equation \(\sqrt{(x-2)^2 + (y+3)^2} = y - 1\). When they square both sides and simplify, they get \((x-2)^2 = 8(y + 1)\), missing the negative sign. This may lead them to select Choice C \(((x - 2)^2 = 8(y + 1))\).

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when expanding \((1-y)^2\) or combining like terms.

Students might expand \((1-y)^2\) incorrectly as \(1 + y^2\) (forgetting the middle term \(-2y\)) or make sign errors when rearranging terms. These calculation mistakes can lead to coefficients like -16 instead of -8. This causes them to get stuck and guess, or potentially select Choice B \(((x - 2)^2 = -16(y + 1))\).

The Bottom Line:

This problem requires understanding both the geometric definition of a parabola and careful attention to the spatial relationship between focus and directrix to determine orientation. Success depends on translating the definition correctly and maintaining algebraic precision throughout multiple steps.