Find the length of the segment in the y-axis that intersects this sphere

Question

Given a sphere with its center at (-1,2,3) and a radius of 4, and a segment AB being a set of points in which the sphere intersects the y-axis, what is the length of AB?

How do I go about solving problems of this kind. I tried picking points in the y-axis and measuring the distance from the center, but that's not the right approach. I'm aware that there might be a calculus solution for this, but this is supposed to be a beginner question and I was looking for a simple, algebraic solution. Also, I'm self-taught, so please be so kind as to referring me to any piece of theory you sense I might be missing on.

Answer 1

First you have to see which points of the $y$-axis belong to the sphere. This leads to the equation$$\bigl(0-(-1)\bigr)^2+(y-2)^2+(0-3)^2=4^2,$$which is equivalent to$$1+(y-2)^2+9=16.$$Solve it. You will find two solutions $y_1$ and $y_2$. The answer to your question is the distance from $A=(0,y_1,0)$ to $B=(0,y_2,0)$.

Answer 2

Solving the equation for x=0 and z=0, to find the 2 points in the y-axis, the equation yields $2-\sqrt{6}$ and $2+\sqrt{6}$. Take the absolute value of both and sum them: $\left|2-\sqrt{6}\right|+2+\sqrt{6}$. The lenght of AB is $2\sqrt{6}$.