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Difference between revisions of "2015 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5"

(Created page with "== Problem == Let <math>A</math> and <math>B</math> be two points in the plane. Describe the set <math>S</math> of all points in the plane such that for any point <math>P</ma...")
 
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Let <math>A</math> and <math>B</math> be two points in the plane. Describe the set <math>S</math> of all points in the plane such
 
Let <math>A</math> and <math>B</math> be two points in the plane. Describe the set <math>S</math> of all points in the plane such
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that for any point <math>P</math> in <math>S</math> we have <cmath>|PA| = 3|PB|</cmath>.
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that for any point <math>P</math> in <math>S</math> we have <cmath>|PA| = 3|PB| \text{.}</cmath>
  
 
== Solution==
 
== Solution==
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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.
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~Puck_0
  
 
== See also ==
 
== See also ==

Latest revision as of 13:29, 21 January 2024

Problem

Let $A$ and $B$ be two points in the plane. Describe the set $S$ of all points in the plane such that for any point $P$ in $S$ we have \[|PA| = 3|PB| \text{.}\]

Solution

WLOG, let $A = (0, 0)$, and $B = (b, 0)$. That means that we have that for any point $(x, y) \in S$, $\sqrt{x^2 + y^2} = 3 \sqrt{(x - b)^2 + y^2} \implies x^2 + y^2 = 9((x - b)^2 + y^2) \implies x^2 + y^2 = 9x^2 - 18xb + 9b^2 + 9y^2 \implies$ $8x^2 - 18xb + 9b^2 + 8y^2 = 0$. Conic sections written in the form $ax^2 + bx + cy^2 + dy + e = 0$ are circles if and only if $a = c$, 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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