Difference between revisions of "2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5"
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== Solution== | == Solution== | ||
− | + | WLOG, let <math>A = (0, 0)</math>, and <math>B = (b, 0)</math>. That means that we have that for any point <math>(x, y) \in S</math>, <math>\sqrt{x^2 + y^2} = 3 \sqrt{(x - b)^2 + y^2} \implies x^2 + y^2 = 9((x - b)^2 + y^2) \implies x^2 + y^2 = 9x^2 - 18xb + 9b^2 + 9y^2 \implies</math> <math>8x^2 - 18xb + 9b^2 + 8y^2 = 0</math>. Conic sections written in the form <math>ax^2 + bx + cy^2 + dy + e = 0</math> are circles if and only if <math>a = c</math>, which is true in our equation. Therefore, S is a circle. | |
− | + | ~Puck_0 | |
== See also == | == See also == |
Revision as of 13:28, 21 January 2024
Problem
Let and be two points in the plane. Describe the set of all points in the plane such that for any point in we have .
Solution
WLOG, let , and . That means that we have that for any point , . Conic sections written in the form are circles if and only if , which is true in our equation. Therefore, S is a circle. ~Puck_0
See also
2015 UNM-PNM Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNM-PNM Problems and Solutions |
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