find the equation o a circle with center (-6,7), tangent at the y-axis

To find the equation of a circle with center \((-6, 7)\) and tangent to the y-axis, we need to follow these steps:

### 1. General Equation of a Circle
The general equation of a circle with center \((h, k)\) and radius \(r\) is:

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

Given:
- Center \((h, k) = (-6, 7)\), so the equation becomes:
  \[
  (x + 6)^2 + (y - 7)^2 = r^2
  \]

### 2. Condition for Tangency to the Y-Axis
A circle is tangent to the y-axis when its radius \(r\) is the horizontal distance between the center of the circle and the y-axis. The distance between the center and the y-axis is the absolute value of the x-coordinate of the center, which is \(|-6| = 6\).

Thus, the radius \(r = 6\).

### 3. Final Equation
Substituting \(r = 6\) into the equation of the circle:

\[
(x + 6)^2 + (y - 7)^2 = 6^2
\]
\[
(x + 6)^2 + (y - 7)^2 = 36
\]

Therefore, the equation of the circle is:

\[
(x + 6)^2 + (y - 7)^2 = 36
\]

Would you like more details or have any questions?

Here are five related questions:
1. How do you find the radius of a circle if only tangency information is provided?
2. What is the geometric interpretation of a circle's tangency to a line?
3. How do you find the equation of a circle when it is tangent to the x-axis instead of the y-axis?
4. Can you describe the effect of shifting the center of a circle on its equation?
5. How would the equation change if the circle were tangent to a different vertical line?

**Tip:** The distance from a point to a line can help determine the radius in tangency problems!

Find the equation of a circle with center (-6,7), tangent at the y-axis.

Learn how to find the equation of a circle with center (-6, 7) that is tangent to the y-axis. This problem covers key concepts in geometry, including the general equation of a circle, tangency conditions, and calculating the radius using the distance formula. Suitable for students in Grades 9-11.

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