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.

gnistory Jun 21, 2024

#1**+1 **

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! :)

NotThatSmart Jun 21, 2024