The polar of line point (2, 1) with respect to the parabola `y^(2)=6x,` is

Options

A. `y=3x+2`
B. `y=3x+6`
C. `3y=x+6`
D. `y=3x+4`

Explanation

To find the polar of the point (2, 1) with respect to the parabola \( y^2 = 6x \), we can follow these steps:

### Step 1: Identify the parameters of the parabola
The given parabola is \( y^2 = 6x \). We can express it in the standard form \( y^2 = 4ax \). Here, we can see that \( 4a = 6 \), which gives us:
\[ a = \frac{6}{4} = \frac{3}{2} \]

### Step 2: Use the polar equation formula
The formula for the polar of a point \((x_1, y_1)\) with respect to the parabola \( y^2 = 4ax \) is given by:
\[ yy_1 = 2a(x + x_1) \]
where \( (x_1, y_1) \) is the point and \( a \) is the parameter we found in Step 1.

### Step 3: Substitute the values into the polar equation
We have the point \( (x_1, y_1) = (2, 1) \) and \( a = \frac{3}{2} \). Substituting these values into the polar equation:
\[ y \cdot 1 = 2 \cdot \frac{3}{2}(x + 2) \]
This simplifies to:
\[ y = 3(x + 2) \]

### Step 4: Expand and rearrange the equation
Now we expand the equation:
\[ y = 3x + 6 \]

### Final Result
Thus, the polar of the point (2, 1) with respect to the parabola \( y^2 = 6x \) is:
\[ y = 3x + 6 \]