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.