Difference between revisions of "2022 SSMO Speed Round Problems/Problem 8"
(Created page with "==Problem== Circle <math>\omega</math> has chord <math>AB</math> of length <math>18</math>. Point <math>X</math> lies on chord <math>AB</math> such that <math>AX = 4.</math> C...") |
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Line 8: | Line 8: | ||
By the Pythagorean Theorem, we have | By the Pythagorean Theorem, we have | ||
− | + | <cmath>(O_1X+CO)^2+(XC)^2 = (OO_1)^2</cmath> | |
− | (O_1X+CO)^2+(XC)^2 | + | <cmath>(O_2X-CO)^2+(XC)^2 = (OO_2)^2.</cmath> |
− | (O_2X-CO)^2+(XC)^2 | ||
− | |||
which is the same as | which is the same as | ||
− | + | <cmath>(r_1+x)^2+25 = (r-r_1)^2</cmath> | |
− | (r_1+x)^2+25 = (r-r_1)^2 | + | <cmath>(r_2-x)^2+25 =(r-r_2)^2.</cmath> |
− | (r_2-x)^2+25 =(r-r_2)^2. | + | \ |
− | \ | ||
− | |||
Solving for <math>r_1</math> and <math>r_2,</math> we have that | Solving for <math>r_1</math> and <math>r_2,</math> we have that | ||
− | + | <cmath>r_1 = \frac{r^2-x^2-25}{2(r+x)}</cmath> | |
− | + | <cmath>r_2 = \frac{r^2-x^2-25}{2(r-x)}.</cmath> | |
− | |||
− | |||
Thus, | Thus, | ||
<cmath> | <cmath> | ||
Line 29: | Line 23: | ||
meaning that the minimum and maximum value of <math>r_1r_2</math> are both <math>\frac{784}{81}</math> so the answer is <math>\boxed{1649}.</math> | meaning that the minimum and maximum value of <math>r_1r_2</math> are both <math>\frac{784}{81}</math> so the answer is <math>\boxed{1649}.</math> | ||
− | + | <center> | |
− | + | <asy> | |
size(7cm); | size(7cm); | ||
point a, b, c, x, o, t, o1, o2; | point a, b, c, x, o, t, o1, o2; | ||
Line 57: | Line 51: | ||
filldraw(circle(o2, abs(o2-x)), opacity(0.2)+palered, lightred); | filldraw(circle(o2, abs(o2-x)), opacity(0.2)+palered, lightred); | ||
− | dot(" | + | dot("$A$", a, dir(145)); |
− | dot(" | + | dot("$B$", b, dir(30)); |
− | dot(" | + | dot("$C$", c, dir(90)); |
− | dot(" | + | dot("$X$", x, dir(60)); |
− | dot(" | + | dot("$O$", o, dir(45)); |
− | dot(" | + | dot("$O_1$", o1); |
− | dot(" | + | dot("$O_2$", o2); |
− | + | </asy> | |
− | + | </center> |
Revision as of 14:03, 3 July 2023
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 which is the same as \ Solving for and we have that Thus, meaning that the minimum and maximum value of are both so the answer is
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("$A$", a, dir(145)); dot("$B$", b, dir(30)); dot("$C$", c, dir(90)); dot("$X$", x, dir(60)); dot("$O$", o, dir(45)); dot("$O_1$", o1); dot("$O_2$", o2); (Error making remote request. Unknown error_msg)