+819 Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19.$ Point $P$ is on segment $AB$ with $AP = 7$, and $Q$ is on segment $CD$ with $CQ = 8$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 25$, find $XY$.
+128707 Let XQ = x and let PY = y
By the interesecting chord theorem we have two equations
PQ * QC = XQ * QY
and
AP * PB = PY * XP
So....subbing we have
11*8 = x * (25 + y)
7*4 = y * (25 + x) simplify these
88 = 25x + xy (1)
28 = 25y + xy (2)
Subtract (2) from (1)
60 = 25 ( x - y)
60 / 25 = x - y
12/5 = x - y
2.4 + y = x
28 = y * ( 25 + 2.4 + y)
28 = y * ( 27.4 + y)
28 = 27.4y + y^2
y^2 + 27.4y - 28 = 0
Solving this for positive y we have that y ≈ .986
And x = 2.4 + .986 ≈ 3.386
XY ≈ .986 + 25 + 3.386 ≈ 29.372
