The equation of the parabola whose focus is (3,-4) and direction `6x-7y+5=0` is

Options

A. `(7x+6y)^(2)-570x+750y+2100=0`
B. `(7x+6y)^(2)+570x-750y+2100=0`
C. `(7x-6y)^(2)-570x+750y+2100=0`
D. `(7x-6y)^(2)+570x-750y+2100=0`

Explanation

To find the equation of the parabola with a focus at (3, -4) and a directrix given by the line \(6x - 7y + 5 = 0\), we will follow these steps:

### Step 1: Identify the focus and the directrix
The focus of the parabola is given as \(F(3, -4)\). The equation of the directrix is \(6x - 7y + 5 = 0\).

### Step 2: Find the distance from a point \((x, y)\) to the focus
The distance \(d_F\) from a point \((x, y)\) to the focus \((3, -4)\) can be calculated using the distance formula:
\[ d_F = \sqrt{(x - 3)^2 + (y + 4)^2} \]

### Step 3: Find the distance from a point \((x, y)\) to the directrix
The distance \(d_D\) from the point \((x, y)\) to the line \(6x - 7y + 5 = 0\) can be calculated using the formula for the distance from a point to a line:
\[ d_D = \frac{|6x - 7y + 5|}{\sqrt{6^2 + (-7)^2}} = \frac{|6x - 7y + 5|}{\sqrt{36 + 49}} = \frac{|6x - 7y + 5|}{\sqrt{85}} \]

### Step 4: Set the distances equal
For a point to lie on the parabola, the distance to the focus must equal the distance to the directrix:
\[ \sqrt{(x - 3)^2 + (y + 4)^2} = \frac{|6x - 7y + 5|}{\sqrt{85}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x - 3)^2 + (y + 4)^2 = \frac{(6x - 7y + 5)^2}{85} \]

### Step 6: Multiply through by 85 to eliminate the fraction
\[ 85[(x - 3)^2 + (y + 4)^2] = (6x - 7y + 5)^2 \]

### Step 7: Expand both sides
Expanding the left side:
\[ 85[(x^2 - 6x + 9) + (y^2 + 8y + 16)] = 85(x^2 + y^2 - 6x + 8y + 25) \]
\[ = 85x^2 + 85y^2 - 510x + 680y + 2125 \]

Expanding the right side:
\[ (6x - 7y + 5)^2 = 36x^2 - 84xy + 49y^2 + 60x - 70y + 25 \]

### Step 8: Set the expanded forms equal
Setting both expanded forms equal:
\[ 85x^2 + 85y^2 - 510x + 680y + 2125 = 36x^2 - 84xy + 49y^2 + 60x - 70y + 25 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ (85x^2 - 36x^2) + (85y^2 - 49y^2) + (510x + 60x) + (680y + 70y) + (2125 - 25) = 0 \]
\[ 49x^2 + 36y^2 - 570x + 750y + 2100 = 0 \]

### Final Equation
The equation of the parabola is:
\[ 49x^2 + 36y^2 - 570x + 750y + 2100 = 0 \]