Find the general equation of the circle tangent to the line 3x-4y=4 at the point P(-4,-4) and center on the line x+y+7=0

To find the general equation of the circle tangent to the line \(3x - 4y = 4\) at the point \(P(-4, -4)\) with its center on the line \(x + y + 7 = 0\), we need to follow these steps:

1. **Determine the coordinates of the center**:
   Since the center lies on the line \(x + y + 7 = 0\), let the coordinates of the center be \((h, k)\). Thus, we have:
   \[
   h + k + 7 = 0 \implies k = -h - 7
   \]

2. **Find the radius**:
   The radius is the distance from the center \((h, k)\) to the point of tangency \(P(-4, -4)\).

   Using the distance formula:
   \[
   r = \sqrt{(h + 4)^2 + (k + 4)^2}
   \]
   Substitute \(k = -h - 7\):
   \[
   r = \sqrt{(h + 4)^2 + (-h - 7 + 4)^2} = \sqrt{(h + 4)^2 + (-h - 3)^2}
   \]

3. **Use the tangency condition**:
   The distance from the center \((h, k)\) to the line \(3x - 4y = 4\) must equal the radius \(r\). The distance from a point \((h, k)\) to the line \(Ax + By + C = 0\) is given by:
   \[
   \text{Distance} = \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}}
   \]
   Here, \(A = 3\), \(B = -4\), and \(C = -4\):
   \[
   \frac{|3h - 4k - 4|}{\sqrt{3^2 + (-4)^2}} = r
   \]
   \[
   \frac{|3h - 4k - 4|}{5} = r
   \]

4. **Set the distances equal**:
   Substitute \(k = -h - 7\) and the expression for \(r\):
   \[
   \frac{|3h - 4(-h - 7) - 4|}{5} = \sqrt{(h + 4)^2 + (-h - 3)^2}
   \]
   Simplify inside the absolute value:
   \[
   |3h + 4h + 28 - 4| = |7h + 24|
   \]
   \[
   \frac{|7h + 24|}{5} = \sqrt{(h + 4)^2 + (-h - 3)^2}
   \]

5. **Square both sides to eliminate the square root**:
   \[
   \left(\frac{|7h + 24|}{5}\right)^2 = (h + 4)^2 + (-h - 3)^2
   \]
   \[
   \frac{(7h + 24)^2}{25} = (h + 4)^2 + (h + 3)^2
   \]

6. **Expand and simplify**:
   \[
   \frac{(7h + 24)^2}{25} = h^2 + 8h + 16 + h^2 + 6h + 9
   \]
   \[
   \frac{49h^2 + 336h + 576}{25} = 2h^2 + 14h + 25
   \]

7. **Multiply through by 25** to clear the denominator:
   \[
   49h^2 + 336h + 576 = 50h^2 + 350h + 625
   \]
   \[
   50h^2 - 49h^2 + 350h - 336h + 625 - 576 = 0
   \]
   \[
   h^2 + 14h + 49 = 0
   \]
   \[
   (h + 7)^2 = 0
   \]
   \[
   h = -7
   \]

8. **Find \(k\)**:
   \[
   k = -h - 7 = -(-7) - 7 = 0
   \]

So the center is \((-7, 0)\), and the radius \(r\) is:
\[
r = \sqrt{((-7) + 4)^2 + (0 + 4)^2} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5
\]

9. **Form the equation of the circle**:
   The equation of the circle with center \((-7, 0)\) and radius 5 is:
   \[
   (x + 7)^2 + y^2 = 25
   \]

Thus, the general equation of the circle is:
\[
(x + 7)^2 + y^2 = 25
\]

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you derive the distance formula?
2. What is the general form of the equation of a circle?
3. How do you find the distance between a point and a line?
4. Can you derive the formula for the distance from a point to a line?
5. What are the properties of a circle tangent to a given line?
6. How do you solve quadratic equations by completing the square?
7. How do you find the radius of a circle given its center and a point on the circle?
8. How do you determine the intersection points of a circle and a line?

**Tip**: When working with geometric problems involving lines and circles, always keep in mind the key properties and relationships such as distances and tangency conditions to simplify your calculations.

Learn how to find the general equation of a circle tangent to the line 3x - 4y = 4 at the point P(-4, -4), with its center on the line x + y + 7 = 0. This problem involves geometry, circle equations, and the distance formula. Suitable for high school students.

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