From the focus of the parabola `y^(2)=8x`, tangents are drawn to the circle `(x-6)^(2)+y^(2)=4`. The equation of the circle through the focus and points of contact of the tangents is :

Options

A. `x^(2)+y^(2)+8x-12=0`
B. `x^(2)+y^(2)-8x+12=0`
C. `x^(2)+y^(2)+6x-12=0`
D. `x^(2)+y^(2)-6x+12=0`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the Focus of the Parabola
The given parabola is \( y^2 = 8x \). We can compare this with the standard form \( y^2 = 4ax \).

From the equation \( y^2 = 8x \), we see that \( 4a = 8 \), which gives us \( a = 2 \). The focus of the parabola is located at the point \( (a, 0) = (2, 0) \).

**Hint:** To find the focus of a parabola in the form \( y^2 = 4ax \), remember that the focus is at the point \( (a, 0) \).

### Step 2: Identify the Center and Radius of the Circle
The equation of the circle is given as \( (x - 6)^2 + y^2 = 4 \).

From this equation, we can identify:
- The center of the circle \( C \) is at \( (6, 0) \).
- The radius \( r \) is \( \sqrt{4} = 2 \).

**Hint:** The standard form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.

### Step 3: Draw the Tangents from the Focus to the Circle
We need to draw tangents from the focus \( F(2, 0) \) to the circle. The points of contact of these tangents will be denoted as \( A \) and \( B \).

### Step 4: Determine the Circle through Points F, A, C, and B
To find the equation of the circle that passes through the points \( F(2, 0) \), \( A \), \( C(6, 0) \), and \( B \), we note that the angle \( \angle AFB \) is \( 90^\circ \) because the tangents from a point outside the circle to the circle are equal in length and subtend a right angle at the point from which they are drawn.

Since \( FC \) acts as a diameter of the circle we are looking for, we can find the midpoint of \( FC \) to determine the center of the new circle.

### Step 5: Find the Midpoint of FC
The midpoint \( M \) of the line segment \( FC \) can be calculated as:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 6}{2}, \frac{0 + 0}{2} \right) = (4, 0) \]

### Step 6: Calculate the Radius of the Circle
The radius \( R \) of the circle can be calculated as half the distance between points \( F \) and \( C \):
\[ FC = \sqrt{(6 - 2)^2 + (0 - 0)^2} = \sqrt{4^2} = 4 \]
Thus, the radius \( R = \frac{4}{2} = 2 \).

### Step 7: Write the Equation of the Circle
Now that we have the center \( (4, 0) \) and radius \( 2 \), we can write the equation of the circle:
\[ (x - 4)^2 + (y - 0)^2 = 2^2 \]
This simplifies to:
\[ (x - 4)^2 + y^2 = 4 \]

### Step 8: Expand the Equation
Expanding the equation gives:
\[ (x^2 - 8x + 16) + y^2 = 4 \]
\[ x^2 + y^2 - 8x + 12 = 0 \]

### Final Answer
The equation of the circle through the focus and points of contact of the tangents is:
\[ x^2 + y^2 - 8x + 12 = 0 \]

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