Find the equation of the parabola whose focus is `(-1,-2)` and the equation of the directrix is given by `4x-3y+2=0`. Also find the equation of the axis.

Answer

To find the equation of the parabola with a focus at (-1, -2) and a directrix given by the equation \(4x - 3y + 2 = 0\), we will follow these steps:

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

### Step 2: Write the general point on the parabola
Let \(P(x, y)\) be a general point on the parabola.

### Step 3: Calculate the distance from the point to the focus
Using the distance formula, the distance \(D\) from point \(P(x, y)\) to the focus \(F(-1, -2)\) is given by:
\[ D = \sqrt{(x + 1)^2 + (y + 2)^2} \]

### Step 4: Calculate the distance from the point to the directrix
The distance \(d\) from point \(P(x, y)\) to the directrix can be calculated using the formula for the distance from a point to a line. For the line \(Ax + By + C = 0\), the distance from point \((x_1, y_1)\) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = 4\), \(B = -3\), and \(C = 2\). Thus, the distance \(d\) is:
\[ d = \frac{|4x - 3y + 2|}{\sqrt{4^2 + (-3)^2}} = \frac{|4x - 3y + 2|}{5} \]

### Step 5: Set the distances equal
For a parabola, the distance from the point to the focus is equal to the distance from the point to the directrix:
\[ \sqrt{(x + 1)^2 + (y + 2)^2} = \frac{|4x - 3y + 2|}{5} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x + 1)^2 + (y + 2)^2 = \frac{(4x - 3y + 2)^2}{25} \]

### Step 7: Multiply through by 25
To eliminate the fraction, multiply both sides by 25:
\[ 25[(x + 1)^2 + (y + 2)^2] = (4x - 3y + 2)^2 \]

### Step 8: Expand both sides
Expanding the left side:
\[ 25[(x^2 + 2x + 1) + (y^2 + 4y + 4)] = 25x^2 + 50x + 25 + 25y^2 + 100y \]
Expanding the right side:
\[ (4x - 3y + 2)^2 = 16x^2 - 24xy + 9y^2 + 16x - 12y + 4 \]

### Step 9: Set the equation to standard form
Combine and rearrange the equation:
\[ 25x^2 + 25y^2 + 50x + 100y + 25 = 16x^2 - 24xy + 9y^2 + 16x - 12y + 4 \]
Rearranging gives:
\[ (25x^2 - 16x^2) + (25y^2 - 9y^2) + (50x - 16x) + (100y + 12y) + (25 - 4) = 0 \]
This simplifies to:
\[ 9x^2 + 16y^2 + 34x + 112y + 21 = 0 \]

### Step 10: Find the equation of the axis
The axis of the parabola is perpendicular to the directrix. The slope of the directrix can be found from its equation. Rearranging \(4x - 3y + 2 = 0\) gives:
\[ y = \frac{4}{3}x + \frac{2}{3} \]
The slope \(m_d\) of the directrix is \(\frac{4}{3}\). Therefore, the slope \(m_a\) of the axis is:
\[ m_a = -\frac{1}{m_d} = -\frac{3}{4} \]
Using the point-slope form of the line through the focus \((-1, -2)\):
\[ y + 2 = -\frac{3}{4}(x + 1) \]
Rearranging gives:
\[ 3x + 4y + 11 = 0 \]

### Final Result
The equation of the parabola is:
\[ 9x^2 + 16y^2 + 34x + 112y + 21 = 0 \]
The equation of the axis is:
\[ 3x + 4y + 11 = 0 \]