persamaan garis singgung lingkaran x² + y² + 4x - 6y - 12 = 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 need to first rewrite the equation of the circle in its standard form and then use the condition of tangency.

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

\[
x^2 + y^2 + 4x - 6y - 12 = 0
\]

We need to rewrite this in standard form by completing the square for \(x\) and \(y\).

#### Completing the Square

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

   \[
   (x^2 + 4x) + (y^2 - 6y) = 12
   \]

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:**

   \[
   (x + 2)^2 - 4 + (y - 3)^2 - 9 = 12
   \]

   \[
   (x + 2)^2 + (y - 3)^2 = 25
   \]

So, the circle is centered at \((-2, 3)\) with a radius \(r = \sqrt{25} = 5\).

### 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 \((-5, 7)\).

#### General Equation of the Tangent Line

The equation of the tangent line to a circle with center \((h, k)\) and radius \(r\) passing through a point \((x_1, y_1)\) is:

\[
(x_1 + 2)x + (y_1 - 3)y = (-2)^2 + (3)^2 + 25
\]

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

Substitute the values into the formula:

\[
(-5 + 2)x + (7 - 3)y = (-2)^2 + (3)^2 + 25
\]

Simplify the equation:

\[
(-3)x + 4y = 4 + 9 + 25
\]

\[
-3x + 4y = 38
\]

Or, equivalently:

\[
3x - 4y + 38 = 0
\]

### Conclusion
The equation of the tangent line to the circle \(x^2 + y^2 + 4x - 6y - 12 = 0\) that passes through the point \((-5, 7)\) is:

\[
\boxed{3x - 4y + 38 = 0}
\]

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

### Further Related Questions:

1. How do you derive the equation of a circle given a center and radius?
2. What are the conditions for a line to be tangent to a circle?
3. How do you find the point of tangency for a given tangent line to a circle?
4. What is the distance from a point to a circle?
5. How can you derive the perpendicular distance formula from a point to a line?

### Tip:
When determining the equation of a tangent line to a circle, always ensure to confirm the point is indeed on the line by substituting back into the equation of the line to check for consistency.

Learn how to find the equation of the tangent line to the circle x² + y² + 4x - 6y - 12 = 0 passing through the point (-5, 7). This problem involves solving for the tangent line using circle equations and conditions for tangency. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation