+116 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.
+1804 Let's create a system of equations to solve this question.
First, we know that the two points must be on the equation \(x^2+y^2=25\)
We also know that x must be 4.
Thus, plugging in x=4 into the first equation, we can find y values. We have
\(4^2+y^2=25\\ y^2=25-16\\ y=\pm\sqrt9\\ y=3, y=-3\)
Thus, the two points A and B are \((4, 3), (4, -3)\)
Since they share an x value, the difference between the y values is the distance. We have
\(3-(-3)=6\)
Thus, 6 is our final answer.
Thanks! :)
