If P(-3, 2) is one end of focal chord PQ of the parabola `y^(2)+ 4x + 4y = 0` then slope of the normal at Q is

Options

A. `-(1)/(2)`
B. 2
C. `(1)/(2)`
D. `-2`

Answer

To find the slope of the normal at point Q of the focal chord PQ of the parabola given by the equation \( y^2 + 4x + 4y = 0 \), we will follow these steps:

### Step 1: Rewrite the Parabola in Standard Form
The given equation is:
\[ y^2 + 4x + 4y = 0 \]
We can rearrange this equation:
\[ y^2 + 4y + 4x = 0 \]
Completing the square for the \( y \) terms:
\[ (y + 2)^2 - 4 + 4x = 0 \]
This simplifies to:
\[ (y + 2)^2 = -4x + 4 \]
or
\[ (y + 2)^2 = -4(x - 1) \]
This is the standard form of a parabola that opens to the left, with vertex at \( (1, -2) \).

### Step 2: Identify the Focus
From the standard form \( (y - k)^2 = -4p(x - h) \), we can identify the vertex \( (h, k) = (1, -2) \) and \( p = 1 \). The focus of the parabola is at:
\[ (h - p, k) = (1 - 1, -2) = (0, -2) \]

### Step 3: Find the Equation of the Focal Chord
We know one end of the focal chord \( P(-3, 2) \). The focal chord's endpoints satisfy the property that the product of their distances from the focus is constant. The coordinates of point \( Q \) can be expressed as \( (x_Q, y_Q) \).

Using the property of focal chords, if \( P(-3, 2) \) is one endpoint, we can find the other endpoint \( Q \) by using the relationship between the coordinates of the endpoints of a focal chord. The equation of the focal chord through \( P \) can be derived as follows:
The slope of the line through points \( P \) and \( Q \) is given by:
\[ \text{slope} = \frac{y_Q - 2}{x_Q + 3} \]
Using the property of focal chords, we can write:
\[ \frac{y + 2 - 2}{x + 3} = -\frac{4}{3} \]
This simplifies to:
\[ y + 2 - 2 = -\frac{4}{3}(x + 3) \]
Thus, the equation of the focal chord becomes:
\[ y + 2 = -\frac{4}{3}(x + 3) \]

### Step 4: Substitute into the Parabola Equation
Substituting \( y \) from the focal chord equation into the parabola's equation:
\[ y = -\frac{4}{3}(x + 3) - 2 \]
Substituting this into the parabola equation \( (y + 2)^2 = -4(x - 1) \):
\[ \left(-\frac{4}{3}(x + 3)\right)^2 = -4(x - 1) \]
Solving this will yield the coordinates of point \( Q \).

### Step 5: Find the Slope of the Tangent at Point Q
To find the slope of the tangent at point \( Q \), we need to differentiate the parabola equation:
\[ \frac{dy}{dx} = \frac{-2}{4} = -\frac{1}{2} \]
This gives us the slope of the tangent line at point \( Q \).

### Step 6: Find the Slope of the Normal
The slope of the normal is the negative reciprocal of the slope of the tangent:
\[ \text{slope of normal} = -\frac{1}{\left(-\frac{1}{2}\right)} = 2 \]

### Final Answer
Thus, the slope of the normal at point \( Q \) is:
\[ \boxed{2} \]