Find the equation of the parabola having focus at(-1,-2) and directrix is x – 2y+3=0.

Answer

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

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

### Step 2: Write the equation of the directrix in slope-intercept form
We can rewrite the directrix equation in slope-intercept form \(y = mx + b\):
\[ x - 2y + 3 = 0 \implies 2y = x + 3 \implies y = \frac{1}{2}x + \frac{3}{2} \]
This shows that the slope \(m = \frac{1}{2}\) and the y-intercept is \(\frac{3}{2}\).

### Step 3: Find the distance from the focus to the directrix
The distance \(d\) from the point \(S(-1, -2)\) to the directrix can be calculated using the formula for the distance from a point to a line \(Ax + By + C = 0\):
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = 1\), \(B = -2\), and \(C = 3\). Thus, substituting \(x_1 = -1\) and \(y_1 = -2\):
\[ d = \frac{|1(-1) - 2(-2) + 3|}{\sqrt{1^2 + (-2)^2}} = \frac{|-1 + 4 + 3|}{\sqrt{1 + 4}} = \frac{|6|}{\sqrt{5}} = \frac{6}{\sqrt{5}} \]

### Step 4: Set up the equation of the parabola
For any point \(P(x, y)\) on the parabola, the distance from \(P\) to the focus \(S\) is equal to the distance from \(P\) to the directrix. This gives us the equation:
\[ \sqrt{(x + 1)^2 + (y + 2)^2} = \frac{|x - 2y + 3|}{\sqrt{5}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides:
\[ (x + 1)^2 + (y + 2)^2 = \frac{(x - 2y + 3)^2}{5} \]

### Step 6: Multiply through by 5 to eliminate the fraction
\[ 5[(x + 1)^2 + (y + 2)^2] = (x - 2y + 3)^2 \]

### Step 7: Expand both sides
Expanding the left side:
\[ 5[(x^2 + 2x + 1) + (y^2 + 4y + 4)] = 5x^2 + 10x + 5 + 5y^2 + 20y + 20 = 5x^2 + 5y^2 + 10x + 20y + 25 \]
Expanding the right side:
\[ (x - 2y + 3)^2 = x^2 - 4xy + 4y^2 + 6x - 12y + 9 \]

### Step 8: Set the equation to zero
Now we set both sides equal to each other:
\[ 5x^2 + 5y^2 + 10x + 20y + 25 = x^2 - 4xy + 4y^2 + 6x - 12y + 9 \]
Rearranging gives us:
\[ (5x^2 - x^2) + (5y^2 - 4y^2) + (10x - 6x) + (20y + 12y) + (25 - 9) + 4xy = 0 \]
This simplifies to:
\[ 4x^2 + y^2 + 4xy + 4x + 32y + 16 = 0 \]

### Final Equation
The equation of the parabola is:
\[ 4x^2 + y^2 + 4xy + 4x + 32y + 16 = 0 \]