The equation of the tangent to the parabola `y^(2)=4x` at the end of the latus rectum in the fourth quadrant is

Options

A. x-y+2 =0
B. x - y + 4=0
C. x+y+2=0
D. x+y+4=0

Answer

To find the equation of the tangent to the parabola \( y^2 = 4x \) at the end of the latus rectum in the fourth quadrant, we can follow these steps:

### Step 1: Identify the parameters of the parabola
The given parabola is \( y^2 = 4x \). This is in the standard form \( y^2 = 4ax \), where \( a \) is the distance from the vertex to the focus.

**Hint**: Compare the given equation with the standard form to find the value of \( a \).

### Step 2: Calculate the value of \( a \)
From the equation \( y^2 = 4x \), we can see that \( 4a = 4 \). Thus, we can find:
\[ a = 1 \]

**Hint**: Remember that the latus rectum of a parabola is defined as the line segment perpendicular to the axis of symmetry that passes through the focus.

### Step 3: Determine the coordinates of the ends of the latus rectum
The coordinates of the ends of the latus rectum for the parabola \( y^2 = 4ax \) are given by:
- \( (a, 2a) \)
- \( (a, -2a) \)

Substituting \( a = 1 \):
- The coordinates are \( (1, 2) \) and \( (1, -2) \).

**Hint**: Identify which point lies in the fourth quadrant.

### Step 4: Identify the point in the fourth quadrant
The point \( (1, -2) \) is in the fourth quadrant.

**Hint**: Recall that the fourth quadrant has positive \( x \) values and negative \( y \) values.

### Step 5: Find the slope of the tangent line at the point
To find the slope of the tangent line at the point \( (1, -2) \), we differentiate the equation of the parabola:
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4x) \]
This gives:
\[ 2y \frac{dy}{dx} = 4 \implies \frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y} \]

**Hint**: Substitute the \( y \)-coordinate of the point to find the slope.

### Step 6: Calculate the slope at the point \( (1, -2) \)
Substituting \( y = -2 \):
\[ \frac{dy}{dx} = \frac{2}{-2} = -1 \]

**Hint**: The slope of the tangent line is crucial for writing the equation of the line.

### Step 7: Use the point-slope form to write the equation of the tangent
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (1, -2) \) and \( m = -1 \):
\[ y - (-2) = -1(x - 1) \]
This simplifies to:
\[ y + 2 = -x + 1 \]

**Hint**: Rearranging the equation will help you find the standard form.

### Step 8: Rearrange the equation
Rearranging gives:
\[ x + y + 1 = 0 \]

Thus, the equation of the tangent to the parabola \( y^2 = 4x \) at the end of the latus rectum in the fourth quadrant is:
\[ \boxed{x + y + 1 = 0} \]