2022 SSMO Speed Round Problems/Problem 8
Problem
Circle has chord
of length
. Point
lies on chord
such that
Circle
with radius
and
with radius
lie on two different sides of
Both
and
are tangent to
at
and
If the sum of the maximum and minimum values of
is
find
.
Solution
Let be the radius of
and let
be the midpoint of
and let
Note that
. WLOG assume that
Since and
we have
By the Pythagorean Theorem, we have
Solving for and
we have that
\begin{align*}
r_1 &= \frac{r^2-x^2-25}{2(r+x)} \
r_2 &= \frac{r^2-x^2-25}{2(r-x)}.
\end{align*}
Thus,
meaning that the minimum and maximum value of
are both
so the answer is
\begin{center} \begin{asy}
size(7cm); point a, b, c, x, o, t, o1, o2; a = (0,0); b = (18,0); c = (9,0); x = (4,0); o = (9, -3);
circle cir = circle(o, abs(a-o)); t = intersectionpoints(cir, line(x,o))[1];
point[] o1o2 = intersectionpoints(ellipse(x, o, (x+t)/2), line(x, x+(0,1)));
o1 = o1o2[0]; o2 = o1o2[1];
draw(o1--o2, red); draw(a--b, blue); draw(c--o, blue); filldraw(cir, opacity(0.2)+lightcyan, blue); // draw(ellipse(x, o, (x+t)/2));
filldraw(circle(o1, abs(o1-x)), opacity(0.2)+palered, lightred); filldraw(circle(o2, abs(o2-x)), opacity(0.2)+palered, lightred);
dot("", a, dir(145)); dot("
", b, dir(30)); dot("
", c, dir(90)); dot("
", x, dir(60)); dot("
", o, dir(45)); dot("
", o1); dot("
", o2);
\end{asy} \end{center}