The equation of a tangent to the parabola, `x^(2) = 8y`, which makes an angle `theta` with the positive direction of x-axis, is:

Options

A. `y=x tan theta + 2cot theta`
B. `y=x tan theta + 2tan theta`
C. `x = y cot theta + 2 tan theta`
D. `y = x tan theta - 2 cot theta`

Explanation

To find the equation of a tangent to the parabola \( x^2 = 8y \) that makes an angle \( \theta \) with the positive direction of the x-axis, we can follow these steps:

### Step 1: Understand the Equation of the Tangent
The equation of the tangent to the parabola \( x^2 = 8y \) at a point \( (x_1, y_1) \) can be expressed in the form:
\[ xx_1 = 4(y + y_1) \]
where \( (x_1, y_1) \) is the point of tangency.

### Step 2: Determine the Slope
The slope of the tangent line can be expressed as:
\[ \text{slope} = \tan \theta \]
This means that the derivative \( \frac{dy}{dx} \) at the point of tangency is equal to \( \tan \theta \).

### Step 3: Differentiate the Parabola
To find the derivative, differentiate the equation \( x^2 = 8y \):
\[ 2x = 8 \frac{dy}{dx} \]
From this, we can express \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{2x}{8} = \frac{x}{4} \]
Setting this equal to \( \tan \theta \):
\[ \frac{x}{4} = \tan \theta \implies x = 4 \tan \theta \]

### Step 4: Find the Corresponding \( y_1 \)
Now, substitute \( x_1 = 4 \tan \theta \) back into the parabola to find \( y_1 \):
\[ x_1^2 = 8y_1 \implies (4 \tan \theta)^2 = 8y_1 \]
This simplifies to:
\[ 16 \tan^2 \theta = 8y_1 \implies y_1 = 2 \tan^2 \theta \]

### Step 5: Substitute \( x_1 \) and \( y_1 \) into the Tangent Equation
Now substitute \( x_1 \) and \( y_1 \) back into the tangent equation:
\[ x(4 \tan \theta) = 4(y + 2 \tan^2 \theta) \]
This simplifies to:
\[ 4x \tan \theta = 4y + 8 \tan^2 \theta \]
Dividing through by 4 gives:
\[ x \tan \theta = y + 2 \tan^2 \theta \]

### Step 6: Rearrange the Equation
Rearranging gives us the final equation of the tangent:
\[ y = x \tan \theta - 2 \tan^2 \theta \]

### Final Result
Thus, the equation of the tangent to the parabola \( x^2 = 8y \) that makes an angle \( \theta \) with the positive direction of the x-axis is:
\[ y = x \tan \theta - 2 \tan^2 \theta \]