Find the standard form of the equation of the circle having the following​ properties:
Center at the point left parenthesis negative 7 comma 7 right parenthesis
Tangent to the y dash axis

The standard form of the equation of a circle is:

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

where \((h, k)\) is the center of the circle and \(r\) is the radius.

In this case:
- The center of the circle is \((-7, 7)\), so \(h = -7\) and \(k = 7\).
- The circle is tangent to the \(y\)-axis, which means the distance from the center to the \(y\)-axis is equal to the radius \(r\).

Since the distance from the point \((-7, 7)\) to the \(y\)-axis is 7 units (the absolute value of the \(x\)-coordinate of the center), the radius \(r = 7\).

Now, substituting the values into the standard form:

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

Simplifying:

\[
(x + 7)^2 + (y - 7)^2 = 49
\]

So, the standard form of the equation of the circle is:

\[
(x + 7)^2 + (y - 7)^2 = 49
\]

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

Here are 5 related questions to extend your understanding:
1. How do you find the radius of a circle given two points on the circle?
2. What happens to the equation if the circle is tangent to the \(x\)-axis instead?
3. How do you find the distance between a point and a line, such as the \(y\)-axis?
4. Can a circle be tangent to both the \(x\)-axis and \(y\)-axis? How would you find its equation?
5. How does the equation of a circle change if the center is at the origin?

**Tip:** Remember, when a circle is tangent to an axis, the radius is simply the distance from the center to that axis!

Find the standard form of the equation of the circle having the following properties: Center at the point (-7, 7) and tangent to the y-axis.

Explore how to find the equation of a circle with its center at (-7, 7) and tangent to the y-axis. This geometry problem walks you through using the circle equation formula, calculating the radius, and understanding tangents. Ideal for students in Grades 9-11.

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