If a circle passing through the point $$(-1, 0)$$ touches y-axis at $$(0, 2)$$, then the x-intercept of the circle is

To solve this problem, we need to find the x-intercept of a circle that passes through the point (-1, 0) and touches the y-axis at (0, 2). The term "x-intercept" here refers to the length of the chord where the circle intersects the x-axis.

Since the circle touches the y-axis at (0, 2), the tangent at this point is vertical. The radius to the point of tangency is perpendicular to the tangent, so it must be horizontal. Therefore, the center of the circle must have the same y-coordinate as the point (0, 2). Let the center be (h, k). Then k = 2.

The radius is the distance from the center (h, 2) to the point of tangency (0, 2), which is |h - 0| = |h|. So the radius r = |h|.

The circle also passes through (-1, 0). The distance from the center (h, 2) to (-1, 0) must equal the radius |h|. Using the distance formula:

$$\sqrt{(h - (-1))^2 + (2 - 0)^2} = |h|$$

Squaring both sides to eliminate the square root and absolute value (since squaring handles the absolute value as |h|² = h²):

$$(h + 1)^2 + 2^2 = h^2$$

Expanding:

$$h^2 + 2h + 1 + 4 = h^2$$

Simplifying:

$$h^2 + 2h + 5 = h^2$$

Subtracting h² from both sides:

$$2h + 5 = 0$$

Solving for h:

$$2h = -5$$

$$h = -\frac{5}{2}$$

So the center is at $$\left(-\frac{5}{2}, 2\right)$$ and the radius is $$\left|-\frac{5}{2}\right| = \frac{5}{2}$$.

The equation of the circle is:

$$\left(x + \frac{5}{2}\right)^2 + (y - 2)^2 = \left(\frac{5}{2}\right)^2$$

Simplifying the right side:

$$\left(x + \frac{5}{2}\right)^2 + (y - 2)^2 = \frac{25}{4}$$

To find the x-intercepts, set y = 0:

$$\left(x + \frac{5}{2}\right)^2 + (0 - 2)^2 = \frac{25}{4}$$

$$(x + \frac{5}{2})^2 + 4 = \frac{25}{4}$$

Converting 4 to a fraction with denominator 4:

$$(x + \frac{5}{2})^2 + \frac{16}{4} = \frac{25}{4}$$

Subtracting $$\frac{16}{4}$$ from both sides:

$$(x + \frac{5}{2})^2 = \frac{25}{4} - \frac{16}{4} = \frac{9}{4}$$

Taking square roots:

$$x + \frac{5}{2} = \pm \frac{3}{2}$$

Solving for x:

First case:

$$x + \frac{5}{2} = \frac{3}{2}$$

$$x = \frac{3}{2} - \frac{5}{2} = -\frac{2}{2} = -1$$

Second case:

$$x + \frac{5}{2} = -\frac{3}{2}$$

$$x = -\frac{3}{2} - \frac{5}{2} = -\frac{8}{2} = -4$$

The circle intersects the x-axis at (-1, 0) and (-4, 0). The length of the chord (the x-intercept length) is the distance between these points:

$$|-1 - (-4)| = |3| = 3$$

Hence, the x-intercept length is 3.

So, the answer is 3.