+645 A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.
The line is x = 4, and the equation of the circle is x^2 + y^2 = 25.
+1876 We can solve the system of equations for x and y and then apply the distance formula to it.
We have the system
\(x=4\\ x^2+y^2=25\)
Subsituting x out from the second equation, we have
\(16+y^2=25\\ y^2=9\\ y= \pm 3\)
So the two points where they intersect are at \((4, 3) \text{ and } (4, -3)\)
Now, we simply apply the distance formula to find the two points. We have
\(\sqrt{(4-4)^2+(3-(-3))^2}\\ \sqrt{0+9}\\ \sqrt{9} = 3 \)
So our final answer is 3.
Thanks! :)
