Which of the following line can be tangent to the parabola `y^2=8x ?` `x-y+2=0` (b) `9x-3y+2=0` `x+2y+8=0` (d) `x+3y+12=0`

Options

A. x-y+2=0
B. 9x-3y+2=0
C. x+2y+8=0
D. x+3y+12=0

Explanation

To determine which of the given lines can be tangent to the parabola \( y^2 = 8x \), we will follow these steps:

### Step 1: Identify the standard form of the parabola
The given parabola is \( y^2 = 8x \). This can be rewritten in the standard form \( y^2 = 4ax \), where \( 4a = 8 \). Thus, we find:
\[ a = 2 \]

### Step 2: Write the equation of the tangent line
The equation of the tangent line to the parabola \( y^2 = 4ax \) is given by:
\[ y = mx + \frac{a}{m} \]
Substituting \( a = 2 \):
\[ y = mx + \frac{2}{m} \]

### Step 3: Convert each line equation to slope-intercept form
We will convert each of the given line equations into the form \( y = mx + c \) to identify their slopes.

1. **For the line \( x - y + 2 = 0 \)**:
\[ y = x + 2 \]
Here, the slope \( m = 1 \).

2. **For the line \( 9x - 3y + 2 = 0 \)**:
\[ 3y = 9x + 2 \implies y = 3x + \frac{2}{3} \]
Here, the slope \( m = 3 \).

3. **For the line \( x + 2y + 8 = 0 \)**:
\[ 2y = -x - 8 \implies y = -\frac{1}{2}x - 4 \]
Here, the slope \( m = -\frac{1}{2} \).

4. **For the line \( x + 3y + 12 = 0 \)**:
\[ 3y = -x - 12 \implies y = -\frac{1}{3}x - 4 \]
Here, the slope \( m = -\frac{1}{3} \).

### Step 4: Check if the lines can be tangents
We will check if these slopes can satisfy the tangent equation \( y = mx + \frac{2}{m} \).

1. **For \( m = 1 \)**:
\[ y = 1x + \frac{2}{1} = x + 2 \]
This matches the first line.

2. **For \( m = 3 \)**:
\[ y = 3x + \frac{2}{3} \]
This does not match the second line \( y = 3x + \frac{2}{3} \).

3. **For \( m = -\frac{1}{2} \)**:
\[ y = -\frac{1}{2}x + \frac{2}{-\frac{1}{2}} = -\frac{1}{2}x - 4 \]
This matches the third line.

4. **For \( m = -\frac{1}{3} \)**:
\[ y = -\frac{1}{3}x + \frac{2}{-\frac{1}{3}} = -\frac{1}{3}x - 6 \]
This does not match the fourth line.

### Conclusion
The lines that can be tangent to the parabola \( y^2 = 8x \) are:
- Option (a) \( x - y + 2 = 0 \)
- Option (c) \( x + 2y + 8 = 0 \)