A tangent is drawn to parabola `y^2=8x` which makes angle `theta` with positive direction of x-axis. The equation of tangent is

Options

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

Answer

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

### Step 1: Identify the parameters of the parabola
The given parabola is \( y^2 = 8x \). This can be compared with the standard form of a parabola \( y^2 = 4ax \), where \( 4a = 8 \).

**Calculation:**
\[ 4a = 8 \implies a = 2 \]

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

### Step 3: Substitute the value of \( a \)
Substituting \( a = 2 \) into the tangent equation:
\[ y = mx + \frac{2}{m} \]

### Step 4: Relate the slope \( m \) to the angle \( \theta \)
The slope \( m \) can be expressed in terms of the angle \( \theta \) that the tangent makes with the positive x-axis:
\[ m = \tan(\theta) \]

### Step 5: Substitute \( m \) into the tangent equation
Now, substituting \( m = \tan(\theta) \) into the tangent equation:
\[ y = \tan(\theta)x + \frac{2}{\tan(\theta)} \]

### Step 6: Rewrite the equation using cotangent
We can rewrite \( \frac{2}{\tan(\theta)} \) as \( 2 \cot(\theta) \):
\[ y = \tan(\theta)x + 2 \cot(\theta) \]

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

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