To find the equation of a circle that is centered at $(-9, -\frac{4}{3})$ and is tangent to the y-axis, follow these steps:

Step 1: Identify the center and radius

• The center of the circle is given as $(-9, -\frac{4}{3})$.
• Since the circle is tangent to the y-axis, the radius of the circle is the horizontal distance from the center to the y-axis.

Step 2: Calculate the radius

• The distance from the center to the y-axis is the absolute value of the x-coordinate of the center, which is $|-9| = 9$.

Step 3: Write the equation of the circle

• The general equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• Plugging in the values, we get:

$(x + 9)^2 + \left(y + \frac{4}{3}\right)^2 = 9^2$

$(x + 9)^2 + \left(y + \frac{4}{3}\right)^2 = 81$

This is the equation of the circle with the given center and tangent to the y-axis.

Would you like further details on this problem? Here are some related questions:

1. How do you calculate the distance between two points?
2. What is the significance of a circle being tangent to a line?
3. How can you find the points of intersection between a circle and a line?
4. How does the general equation of a circle change if it is tangent to the x-axis instead?
5. Can you derive the equation of a circle given a different center and radius?

Tip: The distance formula is essential when working with geometric shapes on the coordinate plane, particularly when determining radii and distances between points.