Power of a Point (Hard)

Late AMC 10/12 through early AIME — dense computation.

Power of a Point: core idea

Chains of powers.

Two circles meet at $A,B$ . Through $P$ on line $AB$ , a line meets circles again at $C,D$ . If $PC=4$ , $PD=9$ , $PA=3$ , find $PB$ .

Hint: Equal powers. Answer: 12.

1. Tangent length $6$ ; external secant $4$ . Whole secant?
2. Power of $(3,4)$ w.r.t. $x^2+y^2=9$ .
3. Circle radius $13$ , chord through a point $5$ from center — chord length?
4. Intersecting chords $3$ & $x$ , $4$ & $12$ . Find $x$ .
5. External power $48$ , whole secant $16$ . External segment?
6. Common chord of $x^2+y^2=25$ and $x^2+y^2-6x-8y=0$ . Length?
7. Diameter $10$ , chord from endpoint length $6$ . Distance from center to chord?
8. Unit circle, tangents from $P$ touch at $A,B$ ; $\angle AOB=60^\circ$ at center. $PA$ ?
9. For $P=(1,2)$ on $x^2+y^2=25$ , product of directed secant segments along slope $1$ ?