The circle C touches the y-axis at point A (0,3) and passes through point B (2,7). Find an equation for C

To find the equation of the circle \( C \) that touches the y-axis at point \( A(0, 3) \) and passes through point \( B(2, 7) \), we can follow these steps:

### Step 1: Determine the center and radius of the circle
The general equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle, and \(r\) is the radius.

### Step 2: Use the condition that the circle touches the y-axis
Since the circle touches the y-axis at point \( A(0, 3) \), the x-coordinate of the center must be the radius \( r \). Therefore, we can express the center of the circle as:
\[
(h, k) = (r, 3)
\]

### Step 3: Use the fact that the circle passes through point \( B(2, 7) \)
The point \( B(2, 7) \) lies on the circle, so it satisfies the equation of the circle. Substituting \( B(2, 7) \) into the circle's equation:
\[
(2 - r)^2 + (7 - 3)^2 = r^2
\]

Simplify this equation:
\[
(2 - r)^2 + 4^2 = r^2
\]
\[
(2 - r)^2 + 16 = r^2
\]

### Step 4: Expand and solve for \( r \)
Expand \( (2 - r)^2 \):
\[
(4 - 4r + r^2) + 16 = r^2
\]
Simplify by canceling \( r^2 \) on both sides:
\[
4 - 4r + 16 = 0
\]
\[
20 - 4r = 0
\]
\[
r = 5
\]

### Step 5: Find the center of the circle
Now that we have \( r = 5 \), the center of the circle is:
\[
(h, k) = (5, 3)
\]

### Step 6: Write the equation of the circle
Finally, substitute \( h = 5 \), \( k = 3 \), and \( r = 5 \) into the general equation of the circle:
\[
(x - 5)^2 + (y - 3)^2 = 5^2
\]
\[
(x - 5)^2 + (y - 3)^2 = 25
\]

So, the equation of the circle \( C \) is:
\[
(x - 5)^2 + (y - 3)^2 = 25
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you determine if a line is tangent to a circle?
2. What is the relationship between a circle's radius and its tangent lines?
3. How do you find the distance between a point and the center of a circle?
4. Can a circle have more than one tangent line at a single point?
5. What happens to the equation of a circle if its radius is doubled?

**Tip:** Always check that any point given as lying on the circle satisfies the final equation of the circle.

Learn how to find the equation of a circle that touches the y-axis at point (0, 3) and passes through point (2, 7). This problem involves using the equation of a circle in center-radius form and solving for the circle's center and radius. Suitable for Grades 9-12 students studying geometry.

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