A circle is tangent to the y-axis at y=3 and has one x-intercept at x=1, find the other x-intercept and the equation of the circle.

...And if it's not any trouble, could you please explain how to do get there alongside showing me what to do?

Thanks (:

To find the other x-intercept and the equation of the circle, we can use the concept of tangents to a circle.

First, let's understand the given information:
1. The circle is tangent to the y-axis at y=3. This means the circle touches the y-axis at a single point located at y=3.
2. The circle has one x-intercept at x=1. This means the circle intersects the x-axis at the point (1, 0).

To find the other x-intercept, we need to determine the radius of the circle first. Since the circle is tangent to the y-axis at y=3, the radius is equal to the distance between the center of the circle and the point of tangency on the y-axis (0, 3).

The distance between two points (x1, y1) and (x2, y2) is given by the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, the distance between (0, 3) and (1, 0) will give us the radius of the circle.
Distance = sqrt((1 - 0)^2 + (0 - 3)^2)
Distance = sqrt(1 + 9)
Distance = sqrt(10)

So the radius of the circle is sqrt(10).

Now that we know the radius, we can find the other x-intercept by considering symmetry. Since the circle is centered on the y-axis, the x-coordinate of the center will be the negative of the x-coordinate of the given x-intercept. Therefore, the center of the circle will be (-1, 0).

To find the other x-intercept, we consider that the sum of the x-coordinates of any two points on the circumference of a circle is equal to the negative of the x-coordinate of the center.

Using this information, we can write the equation:
1st x-intercept + 2nd x-intercept = -1

Substituting the known value of the 1st x-intercept (1), we get:
1 + 2nd x-intercept = -1
2nd x-intercept = -2

So the other x-intercept is x = -2.

Now we can write the equation of the circle using the center and the radius.
The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle.

Substituting the values, we have:
(x - (-1))^2 + (y - 0)^2 = (sqrt(10))^2
(x + 1)^2 + y^2 = 10

Therefore, the equation of the circle is (x + 1)^2 + y^2 = 10.

Hope this explanation helps! Let me know if you have any further questions.