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The question in the image asks to determine the equation of the tangent line to the circle given by the equation:

\[
x^2 + y^2 - 2x + 8y + 4 = 0
\]

which is perpendicular to the line \( x - y = 2 \).

### Step 1: Rewrite the Circle Equation
We will first complete the square to express the circle equation in a standard form. The given equation is:

\[
x^2 + y^2 - 2x + 8y + 4 = 0
\]

Group the \(x\)-terms and the \(y\)-terms:

\[
(x^2 - 2x) + (y^2 + 8y) = -4
\]

Now, complete the square for both groups.

For \(x^2 - 2x\), add and subtract \(\left(\frac{2}{2}\right)^2 = 1\):

\[
(x^2 - 2x + 1) - 1 = 0
\]

For \(y^2 + 8y\), add and subtract \(\left(\frac{8}{2}\right)^2 = 16\):

\[
(y^2 + 8y + 16) - 16 = 0
\]

The equation becomes:

\[
(x - 1)^2 - 1 + (y + 4)^2 - 16 = -4
\]

Simplify:

\[
(x - 1)^2 + (y + 4)^2 = 13
\]

This represents a circle with center \((1, -4)\) and radius \(\sqrt{13}\).

### Step 2: Perpendicular Line Slope
The line equation \( x - y = 2 \) can be rewritten in slope-intercept form as:

\[
y = x - 2
\]

The slope of this line is \( m = 1 \). A line that is perpendicular to this must have a slope of \( m = -1 \) (the negative reciprocal of 1).

### Step 3: Equation of Tangent Line
The equation of a line with slope \( -1 \) can be written as:

\[
y = -x + c
\]

To find the value of \(c\), we use the fact that the line is tangent to the circle, meaning the distance between the center of the circle \((1, -4)\) and the line \( y = -x + c \) must equal the radius, \( \sqrt{13} \).

The formula for the distance from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is:

\[
d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]

For the line \( y = -x + c \), we rewrite it as \(x + y - c = 0\), so \( A = 1 \), \( B = 1 \), and \( C = -c \).

Substitute the center \((1, -4)\) into the distance formula:

\[
d = \frac{|1(1) + 1(-4) - c|}{\sqrt{1^2 + 1^2}} = \frac{|1 - 4 - c|}{\sqrt{2}} = \frac{| -3 - c|}{\sqrt{2}}
\]

Set this equal to the radius \(\sqrt{13}\):

\[
\frac{| -3 - c|}{\sqrt{2}} = \sqrt{13}
\]

Multiply both sides by \(\sqrt{2}\):

\[
| -3 - c| = \sqrt{26}
\]

This gives two cases:

1. \( -3 - c = \sqrt{26} \), so \( c = -3 - \sqrt{26} \)
2. \( -3 - c = -\sqrt{26} \), so \( c = -3 + \sqrt{26} \)

### Final Answer:
The equations of the two possible tangent lines are:

\[
y = -x - 3 - \sqrt{26}
\]

or

\[
y = -x - 3 + \sqrt{26}
\]

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do you complete the square for quadratic expressions?
2. Why is the distance from a point to a line important in geometry?
3. How is the slope of a perpendicular line calculated?
4. What are some methods to find tangent lines to circles?
5. How do you interpret the solution in the context of geometry?

**Tip:** Always check the discriminant when finding the distance between a point and a line to ensure the line is tangent.

Tentukan persamaan garis singgung lingkaran x^2 + y^2 - 2x + 8y + 4 = 0 yang tegak lurus dengan garis x - y = 2.

Determine the equation of the tangent line to the circle x^2 + y^2 - 2x + 8y + 4 = 0, which is perpendicular to the line x - y = 2. This problem involves completing the square, calculating the slope of perpendicular lines, and using the point-to-line distance formula.

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