Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2016 AMC 12A Problems/Problem 7"
 
(2 intermediate revisions by 2 users not shown)
Line 2: Line 2:
 
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?
 
Which of these describes the graph of <math>x^2(x+y+1)=y^2(x+y+1)</math> ?
  
−
<math>\textbf{(A)}\ \text{two parallel lines}\\ \qquad\textbf{(B)}\ \text{two intersecting lines}\\ \qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\ \qquad\textbf{(D)}\ \text{three lines that do not all pass through a comment point}\\ \qquad\textbf{(E)}\ \text{a line and a parabola}</math>  
+
<math>\textbf{(A)}\ \text{two parallel lines}\\ \textbf{(B)}\ \text{two intersecting lines}\\ \textbf{(C)}\ \text{three lines that all pass through a common point}\\ \textbf{(D)}\ \text{three lines that do not all pass through a common point}\\ \textbf{(E)}\ \text{a line and a parabola}</math>
  
 
==Solution 1==
 
==Solution 1==
Line 10: Line 10:
 
==Solution 2==
 
==Solution 2==
  
−
If <math>x+y+1\neq0</math>, then dividing both sides of the equation by <math>x+y+1</math> gives us <math>x^2=y^2</math>. Rearranging and factoring, we get <math>x^2-y^2=(x+y)(x-y)=0</math>. If <math>x+y+1=0</math>, then the equation is satisfied. Thus either <math>x+y=0</math>, <math>x-y=0</math>, and <math>x+y+1=0</math>. These equations can be rearranged into the lines <math>y=-x</math>, <math>y=x</math>, and <math>y=-x-1</math>, respectively. Since these three lines are distinct, the answer is <math>\boxed{\textbf{(D)}\; \text{three lines that do not all pass through a common point}}</math>.
+
If <math>x+y+1\neq0</math>, then dividing both sides of the equation by <math>x+y+1</math> gives us <math>x^2=y^2</math>. Rearranging and factoring, we get <math>x^2-y^2=(x+y)(x-y)=0</math>. If <math>x+y+1=0</math>, then the equation is satisfied. Thus either <math>x+y=0</math>, <math>x-y=0</math>, or <math>x+y+1=0</math>. These equations can be rearranged into the lines <math>y=-x</math>, <math>y=x</math>, and <math>y=-x-1</math>, respectively. Since these three lines are distinct, the answer is <math>\boxed{\textbf{(D)}\; \text{three lines that do not all pass through a common point}}</math>.
  
 
==Solution 3==
 
==Solution 3==

Latest revision as of 17:32, 26 December 2020

Problem

Which of these describes the graph of $x^2(x+y+1)=y^2(x+y+1)$ ?

$\textbf{(A)}\ \text{two parallel lines}\\ \textbf{(B)}\ \text{two intersecting lines}\\ \textbf{(C)}\ \text{three lines that all pass through a common point}\\ \textbf{(D)}\ \text{three lines that do not all pass through a common point}\\ \textbf{(E)}\ \text{a line and a parabola}$

Solution 1

The equation $x^2(x+y+1)=y^2(x+y+1)$ tells us $x^2=y^2$ or $x+y+1=0$ . $x^2=y^2$ generates two lines $y=x$ and $y=-x$ . $x+y+1=0$ is another straight line. The only intersection of $y=x$ and $y=-x$ is $(0,0)$ , which is not on $x+y+1=0$ . Therefore, the graph is three lines that do not have a common intersection, or $\boxed{\textbf{(D)}\; \text{three lines that do not all pass through a common point}}$

Solution 2

If $x+y+1\neq0$, then dividing both sides of the equation by $x+y+1$ gives us $x^2=y^2$. Rearranging and factoring, we get $x^2-y^2=(x+y)(x-y)=0$. If $x+y+1=0$, then the equation is satisfied. Thus either $x+y=0$, $x-y=0$, or $x+y+1=0$. These equations can be rearranged into the lines $y=-x$, $y=x$, and $y=-x-1$, respectively. Since these three lines are distinct, the answer is $\boxed{\textbf{(D)}\; \text{three lines that do not all pass through a common point}}$.

Solution 3

Subtract $y^2(x+y+1)$ on both sides of the equation to get $x^2(x+y+1)-y^2(x+y+1)=0$. Factoring $x+y+1$ gives us $(x+y+1)(x^2-y^2)=(x+y+1)(x+y)(x-y)=0$, so either $x+y+1=0$, $x+y=0$, or $x-y=0$. Continue on with the second half of solution 2.

Diagram:

$AB: y=x$

$CD: y=-x$

$EF: x+y+1=0$

[asy] size(7cm); pair F= (5,0), E=(-1,6), D=(0,0), C=(6,0), B=(6,6), A=(0,6); draw(A--C); draw(B--D); draw(E--F); label("$A$", A, dir(135)); label("$B$", C, dir(-45)); label("$C$", B, dir(45)); label("$D$", D, dir(-135)); label("$E$", E, dir(135)); label("$F$", F, dir(-45)); [/asy]

See Also

2016 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12A_Problems/Problem_7&oldid=140670"