Find the equation of the parabola with focus at (3, -4) and directrix x + y - 2 = 0 .

Options

A. `x^(2) +4xy + y^(2) -8x + 20y + 46 = 0 `
B. `x^(2) +2xy + y^(2) -8x + 20y + 46 = 0 `
C. `x^(2) - 2xy + y^(2) -8x + 20y + 46 = 0 `
D. `x^(2) - 4xy + y^(2) -8x + 20y + 46 = 0 `

Answer

To find the equation of the parabola with a focus at (3, -4) and a directrix given by the line \(x + y - 2 = 0\), we will follow these steps:

### Step 1: Identify the Focus and Directrix
The focus of the parabola is given as \(F(3, -4)\) and the directrix is the line \(x + y - 2 = 0\).

### Step 2: Use the Definition of a Parabola
A point \(P(h, k)\) on the parabola is defined such that the distance from \(P\) to the focus \(F\) is equal to the perpendicular distance from \(P\) to the directrix.

### Step 3: Calculate the Distance to the Focus
The distance \(PS\) from point \(P(h, k)\) to the focus \(F(3, -4)\) can be calculated using the distance formula:
\[ PS = \sqrt{(h - 3)^2 + (k + 4)^2} \]

### Step 4: Calculate the Distance to the Directrix
The distance \(PM\) from point \(P(h, k)\) to the directrix \(x + y - 2 = 0\) can be calculated using the formula for the distance from a point to a line \(Ax + By + C = 0\):
\[ PM = \frac{|h + k - 2|}{\sqrt{1^2 + 1^2}} = \frac{|h + k - 2|}{\sqrt{2}} \]

### Step 5: Set the Distances Equal
According to the definition of a parabola:
\[ \sqrt{(h - 3)^2 + (k + 4)^2} = \frac{|h + k - 2|}{\sqrt{2}} \]

### Step 6: Square Both Sides
Squaring both sides to eliminate the square root gives:
\[ (h - 3)^2 + (k + 4)^2 = \frac{(h + k - 2)^2}{2} \]

### Step 7: Expand Both Sides
Expanding both sides:
1. Left side:
\[ (h - 3)^2 + (k + 4)^2 = (h^2 - 6h + 9) + (k^2 + 8k + 16) = h^2 + k^2 - 6h + 8k + 25 \]

2. Right side:
\[ \frac{(h + k - 2)^2}{2} = \frac{(h^2 + 2hk + k^2 - 4h - 4k + 4)}{2} = \frac{1}{2}h^2 + hk - 2h - 2k + 2 \]

### Step 8: Set the Expanded Forms Equal
Now we equate the two expanded forms:
\[ h^2 + k^2 - 6h + 8k + 25 = \frac{1}{2}h^2 + hk - 2h - 2k + 2 \]

### Step 9: Rearrange the Equation
Multiply through by 2 to eliminate the fraction:
\[ 2h^2 + 2k^2 - 12h + 16k + 50 = h^2 + 2hk - 4h - 4k + 4 \]
Rearranging gives:
\[ h^2 + 2k^2 - 8h + 20k + 46 = 0 \]

### Step 10: Final Equation
This is the equation of the parabola in standard form.