Which of the following line can be normal to parabola `y^2=12 x ?` `x+y-9=0` (b) `2x-y-32=0` `2x+y-36=0` (d) `3x-y-72=0`

Options

A. x+y-9=0
B. 2x-y-32=0
C. 2x+y-36=0
D. 3x-y-99=0

Explanation

To determine which of the given lines can be normal to the parabola \( y^2 = 12x \), we will follow these steps:

### Step 1: Identify the parameters of the parabola
The standard form of the parabola is given by \( y^2 = 4ax \). Here, we have \( 4a = 12 \), which gives us:
\[ a = \frac{12}{4} = 3 \]

### Step 2: Write the equation of the normal to the parabola
The equation of the normal to the parabola \( y^2 = 4ax \) at a point where the slope of the tangent is \( m \) is given by:
\[ y = mx - 2am - am^3 \]
Substituting \( a = 3 \) into the equation, we get:
\[ y = mx - 6m - 3m^3 \]

### Step 3: Analyze each line to find its slope
We will analyze each line to find its slope \( m \).

1. **Line 1: \( x + y - 9 = 0 \)**
- Rearranging gives: \( y = -x + 9 \)
- Slope \( m = -1 \)

2. **Line 2: \( 2x - y - 32 = 0 \)**
- Rearranging gives: \( y = 2x - 32 \)
- Slope \( m = 2 \)

3. **Line 3: \( 2x + y - 36 = 0 \)**
- Rearranging gives: \( y = -2x + 36 \)
- Slope \( m = -2 \)

4. **Line 4: \( 3x - y - 72 = 0 \)**
- Rearranging gives: \( y = 3x - 72 \)
- Slope \( m = 3 \)

### Step 4: Check which slopes satisfy the normal equation
Now we will substitute each slope into the normal equation \( y = mx - 6m - 3m^3 \) and check if it matches the line's equation.

1. **For \( m = -1 \) (Line 1)**
\[ y = -1x - 6(-1) - 3(-1)^3 = -x + 6 + 3 = -x + 9 \]
This matches Line 1.

2. **For \( m = 2 \) (Line 2)**
\[ y = 2x - 6(2) - 3(2)^3 = 2x - 12 - 24 = 2x - 36 \]
This does not match Line 2.

3. **For \( m = -2 \) (Line 3)**
\[ y = -2x - 6(-2) - 3(-2)^3 = -2x + 12 + 24 = -2x + 36 \]
This matches Line 3.

4. **For \( m = 3 \) (Line 4)**
\[ y = 3x - 6(3) - 3(3)^3 = 3x - 18 - 81 = 3x - 99 \]
This matches Line 4.

### Conclusion
The lines that can be normal to the parabola \( y^2 = 12x \) are:
- Line 1: \( x + y - 9 = 0 \)
- Line 3: \( 2x + y - 36 = 0 \)
- Line 4: \( 3x - y - 72 = 0 \)

### Final Answer
The correct options are (a), (c), and (d).

---