+613 Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{AB}$ is $4,$ and the circle has a radius of $5.$ Find the length $AB.$
+15068 The length of AP is 4, and the circle has a radius of 5. Find the length AB.
fcircle(x)=√52−x2A (0,5)P (4,5)fPB(x)=−√42−(x−4)2+5=−√16−x2+8x−16+5fPB(x)=−√8x−x2+5
√25−x2=−√8x−x2+5√5−x√x+5=5−√8−x√x(WolframAlpha)xB∈{0,20041}yb=√25−4.¯87804 2=1.0976B (20041,√20.¯12195)B (4.¯87804,1.0976)
¯AB=√(yA−yB)2+(xB−xA)2=√(5−1.0976)2+(4.¯87804−0)2¯AB=6.2469
The length ¯AB is 6.2469.
!
