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.

 Jun 21, 2024

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! :)

 Jun 21, 2024

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