Find the equation of parabola whose
focus is (-3,0) and directrix is x + 5 = 0

Answer

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

### Step 1: Identify the focus and directrix
The focus of the parabola is given as \( F(-3, 0) \) and the directrix is the line \( x + 5 = 0 \), which can be rewritten as \( x = -5 \).

### Step 2: Set up the distance formula
A point \( (x, y) \) on the parabola is equidistant from the focus and the directrix. Therefore, we can set up the equation based on the definition of a parabola:
\[ \text{Distance from } (x, y) \text{ to the focus} = \text{Distance from } (x, y) \text{ to the directrix} \]

The distance from the point \( (x, y) \) to the focus \( (-3, 0) \) is:
\[ \sqrt{(x + 3)^2 + (y - 0)^2} = \sqrt{(x + 3)^2 + y^2} \]

The distance from the point \( (x, y) \) to the directrix \( x = -5 \) is:
\[ |x + 5| \]

### Step 3: Set up the equation
Equating the two distances gives us:
\[ \sqrt{(x + 3)^2 + y^2} = |x + 5| \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:
\[ (x + 3)^2 + y^2 = (x + 5)^2 \]

### Step 5: Expand both sides
Expanding both sides, we get:
\[ (x^2 + 6x + 9) + y^2 = (x^2 + 10x + 25) \]

### Step 6: Simplify the equation
Now, we simplify the equation:
\[ x^2 + 6x + 9 + y^2 = x^2 + 10x + 25 \]

Subtract \( x^2 \) from both sides:
\[ 6x + 9 + y^2 = 10x + 25 \]

Rearranging gives:
\[ y^2 = 10x - 6x + 25 - 9 \]
\[ y^2 = 4x + 16 \]

### Step 7: Final equation
We can rewrite this as:
\[ y^2 = 4(x + 4) \]

This is the equation of the parabola in standard form.

### Summary
The equation of the parabola whose focus is at (-3, 0) and directrix is \( x + 5 = 0 \) is:
\[ y^2 = 4(x + 4) \]