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.