Find the equation of the normal to the parabola `y^(2)=4x` which is parallel to `y-2x+5=0`.

Answer

To find the equation of the normal to the parabola \( y^2 = 4x \) that is parallel to the line \( y - 2x + 5 = 0 \), we can follow these steps:

### Step 1: Identify the parameters of the parabola
The given parabola is \( y^2 = 4x \). We can compare this with the standard form \( y^2 = 4ax \) to find \( a \).

\[ 4a = 4 \implies a = 1 \]

**Hint:** The value of \( a \) helps us understand the geometry of the parabola.

### Step 2: Find the slope of the given line
The equation of the line is given as \( y - 2x + 5 = 0 \). We can rearrange this into slope-intercept form \( y = mx + c \):

\[ y = 2x - 5 \]

From this, we see that the slope \( m \) of the line is \( 2 \).

**Hint:** The slope of the normal line must match the slope of the given line.

### Step 3: Write the equation of the normal to the parabola
The equation of the normal to the parabola at a point \( (x_1, y_1) \) can be expressed as:

\[ y = mx - 2am - am^3 \]

Substituting \( a = 1 \) and \( m = 2 \):

\[ y = 2x - 2(1)(2) - (1)(2^3) \]

**Hint:** Substitute the values correctly to find the equation of the normal.

### Step 4: Simplify the equation
Now, let's simplify the equation:

\[ y = 2x - 4 - 8 \]
\[ y = 2x - 12 \]

**Hint:** Make sure to combine like terms correctly.

### Final Result
Thus, the equation of the normal to the parabola \( y^2 = 4x \) which is parallel to the line \( y - 2x + 5 = 0 \) is:

\[ \boxed{y = 2x - 12} \]