Finding the length from a point to a center of the circle.

Question

In the diagram, $AB = 2$ and $BP = 6.$ If the radius of the circle is 11, find the distance from $P$ to the center of the circle.

How can I use power of a point here to find the length of P to the center of the circle? What is the relation of the given values with the length that needs to be found?

Answer 1

All you need is the Pythagorean theorem. If $O$ is the center of the circle, draw the perpendicular to $AB$ from $O$ and let the foot of the perpendicular be point $M$. Then because $AO = BO = 11$, $\triangle AOB$ is isosceles and altitude $OM$ bisects $AB$. So $AM = BM = 1$. Now draw $OP$ to make $\triangle PMO$, which has a right angle at $M$. Then two applications of the Pythagorean theorem solve for $OP$: $$BO^2 = OM^2 + BM^2, \\ OP^2 = PM^2 + BM^2.$$

Answer 2

By using the power of a point formula, we can write

$$PA.PB = OP^2-R^2= \; power \; of \; P$$

thus

$$OP^2=8.6+11^2=169$$

and $$OP=13$$