Q20

 Q20 - In the diagram, circle O has a radius of 10, circle P is internally tangent to O and has a radius of 4. is tangent to circle P at T and, if drawn, line intersects circle O at points A and B. Compute the product TA•TB.


Answer: 60
Solution:


Extend QP and QT so that they intersect circle O at V and S, respectively.
By the Power of a Point Theorem, in the big circle,
Notice that QTS and ATB are chords in the big circle passing through the same point T,
So AT.TB = QT.TS
So if we find QT and TS, we are done.
QV = 20(diameter of O).
QP = QV - 4 = 16.
Triangle QPT is right triangle so QT^2 = QP^2 - PT^2 = 256 - 16.
QT = sqrt(240) = 4.sqrt(15).
Now, QSV = 90 deg cause QV is the diameter subtending the angle at S.
So QVS and QPT are similar.
QP/QV = QT/QS = 4.sqrt(15)/QS = 16/20
QS = 5.sqrt(15)
TS = QS- QT = sqrt(5)
QT.TS = 4.5 = 20. 




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