A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.

The circle is x^2 + y^2 = 1, and the line is y = x.

ABJeIIy Jul 29, 2024

#1**+1 **

We essentially have a system of equations to solve for x and y. The equation of

\( x^2 + y^2 = 1\\ y = x\)

Now, from the second equation, we know that x=y. Thus, subsituting out y for x, we have

\(x^2+x^2=1\\ 2x^2=1\\ x^2=1/2\\ x=\pm \frac{1}{\sqrt2} = \pm\frac{\sqrt2}{2}\)

Since x is equal to y, we have two points. We have

\((\frac{\sqrt2}{2},\frac{\sqrt2}{2}), (-\frac{\sqrt2}{2},-\frac{\sqrt2}{2})\)

Using the distance formula on these two points (or you could just graph it and count), we get

\(\sqrt{\left(-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\right)^2+\left(-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\right)^2}=2\)

So 2 is the answer.

Thanks! :)

NotThatSmart Jul 29, 2024