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.

ABJeIIy Oct 27, 2024

#1**+1 **

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! :)

NotThatSmart Oct 28, 2024