geometry

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! :)