persamaan garis singgung lingkaran x² + y² + 4x - 6y - 1 = 0 yang melalui titik (-5, 7) adalah

To find the equation of the tangent line to the circle that passes through a given point, we first need to understand the equation of the circle and the point through which the tangent line should pass.

### Step 1: Rewrite the Equation of the Circle
The given circle equation is:

\[
x^2 + y^2 + 4x - 6y - 1 = 0
\]

We can rewrite this equation in standard form by completing the square.

#### Completing the Square

1. **Group the \(x\) and \(y\) terms**:

   \[
   (x^2 + 4x) + (y^2 - 6y) = 1
   \]

2. **Complete the square for \(x\) and \(y\):**

   - For \(x^2 + 4x\), take half of 4, square it, and add it inside the parenthesis:
   
   \[
   x^2 + 4x = (x + 2)^2 - 4
   \]

   - For \(y^2 - 6y\), take half of -6, square it, and add it inside the parenthesis:
   
   \[
   y^2 - 6y = (y - 3)^2 - 9
   \]

3. **Rewrite the equation:**

   \[
   (x + 2)^2 - 4 + (y - 3)^2 - 9 = 1
   \]

   \[
   (x + 2)^2 + (y - 3)^2 = 14
   \]

So, the center of the circle is \((-2, 3)\) and the radius \(r = \sqrt{14}\).

### Step 2: Find the Equation of the Tangent Line
We want to find the equation of the tangent line to the circle that passes through the point \((-5, 7)\).

#### Condition for Tangency

The equation of the tangent line to a circle with center \((h, k)\) and radius \(r\) at a point \((x_1, y_1)\) not on the circle is given by:

\[
(x_1 + h)x + (y_1 + k)y = h^2 + k^2 + r^2
\]

Here, \(h = -2\), \(k = 3\), and \((x_1, y_1) = (-5, 7)\).

Substitute the values into the formula:

\[
((-5) + (-2))x + ((7) + 3)y = (-2)^2 + (3)^2 + (\sqrt{14})^2
\]

Simplify the equation:

\[
(-7)x + (10)y = 4 + 9 + 14
\]

\[
-7x + 10y = 27
\]

Or, equivalently:

\[
7x - 10y + 27 = 0
\]

### Conclusion
The equation of the tangent line to the circle \(x^2 + y^2 + 4x - 6y - 1 = 0\) that passes through the point \((-5, 7)\) is:

\[
\boxed{7x - 10y + 27 = 0}
\]

Would you like more details or have any questions about this solution?

### Further Related Questions:

1. What is the length of the tangent line from a given point to a circle?
2. How do you find the points of intersection between a line and a circle?
3. How can you determine if a point is inside, on, or outside a circle?
4. What are the parametric equations for a circle?
5. How do you derive the standard form of a circle from its general form?

### Tip:
When working with circles, always remember to complete the square to convert the general form to the standard form. This makes it easier to identify the circle's center and radius.

Discover how to find the equation of the tangent line to the circle x² + y² + 4x - 6y - 1 = 0 that passes through the point (-5, 7). This problem involves key concepts such as the equation of a circle, tangent lines, and completing the square method. Suitable for advanced high school students.

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