Find the equations of the tangents to the circle `x^(2) + y^(2) -8y - 8 = 0` which are parallel to the line 5 x -2 y = 2.

Answer

To find the equations of the tangents to the circle given by the equation \(x^2 + y^2 - 8y - 8 = 0\) that are parallel to the line \(5x - 2y = 2\), we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we need to rewrite the equation of the circle in standard form. We start with:
\[ x^2 + y^2 - 8y - 8 = 0 \]
We can rearrange this to:
\[ x^2 + (y^2 - 8y) = 8 \]
Now, we complete the square for the \(y\) terms:
\[ y^2 - 8y = (y - 4)^2 - 16 \]
Substituting this back, we have:
\[ x^2 + (y - 4)^2 - 16 = 8 \]
This simplifies to:
\[ x^2 + (y - 4)^2 = 24 \]
Thus, the center of the circle is \((0, 4)\) and the radius \(r\) is \(\sqrt{24} = 2\sqrt{6}\).

### Step 2: Determine the Slope of the Given Line
Next, we need to find the slope of the line \(5x - 2y = 2\). We can rewrite it in slope-intercept form:
\[ -2y = -5x + 2 \quad \Rightarrow \quad y = \frac{5}{2}x - 1 \]
From this, we see that the slope \(m\) is \(\frac{5}{2}\).

### Step 3: Write the Equation of the Tangent Line
Since we want the tangent lines to be parallel to this line, they will also have the same slope. Therefore, we can express the equation of the tangent line in point-slope form:
\[ y - 4 = \frac{5}{2}(x - 0) \quad \Rightarrow \quad y = \frac{5}{2}x + c \]
This can also be written as:
\[ 5x - 2y + c = 0 \]

### Step 4: Use the Distance Formula
The distance from the center of the circle \((0, 4)\) to the line \(5x - 2y + c = 0\) must equal the radius \(2\sqrt{6}\). The distance \(d\) from a point \((h, k)\) to a line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} \]
For our case:
- \(A = 5\)
- \(B = -2\)
- \(C = c\)
- \(h = 0\)
- \(k = 4\)

Substituting these values into the distance formula:
\[ d = \frac{|5(0) - 2(4) + c|}{\sqrt{5^2 + (-2)^2}} = \frac{|c - 8|}{\sqrt{25 + 4}} = \frac{|c - 8|}{\sqrt{29}} \]
Setting this equal to the radius:
\[ \frac{|c - 8|}{\sqrt{29}} = 2\sqrt{6} \]
Multiplying both sides by \(\sqrt{29}\):
\[ |c - 8| = 2\sqrt{6} \cdot \sqrt{29} = 2\sqrt{174} \]

### Step 5: Solve for \(c\)
This gives us two equations:
1. \(c - 8 = 2\sqrt{174}\)
2. \(c - 8 = -2\sqrt{174}\)

Solving these:
1. \(c = 8 + 2\sqrt{174}\)
2. \(c = 8 - 2\sqrt{174}\)

### Step 6: Write the Tangent Equations
Now we can write the equations of the tangents:
1. For \(c = 8 + 2\sqrt{174}\):
\[ 5x - 2y + (8 + 2\sqrt{174}) = 0 \quad \Rightarrow \quad 5x - 2y + 8 + 2\sqrt{174} = 0 \]
2. For \(c = 8 - 2\sqrt{174}\):
\[ 5x - 2y + (8 - 2\sqrt{174}) = 0 \quad \Rightarrow \quad 5x - 2y + 8 - 2\sqrt{174} = 0 \]

Thus, the equations of the tangents are:
\[ 5x - 2y + 8 + 2\sqrt{174} = 0 \quad \text{and} \quad 5x - 2y + 8 - 2\sqrt{174} = 0 \]