Difference between revisions of "2005 AMC 12B Problems/Problem 8"
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== Problem == | == Problem == | ||
| + | For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the | ||
| + | vertex of the parabola <math>y = x^2 + a^2</math> ? | ||
| + | |||
| + | <math> | ||
| + | \mathrm{(A)}\ 0 \qquad | ||
| + | \mathrm{(B)}\ 1 \qquad | ||
| + | \mathrm{(C)}\ 2 \qquad | ||
| + | \mathrm{(D)}\ 10 \qquad | ||
| + | \mathrm{(E)}\ \text{infinitely many} | ||
| + | </math> | ||
== Solution == | == Solution == | ||
| − | + | We see that the vertex of the quadratic function <math>y = x^2 + a^2</math> is <math>(0,\,a^2)</math>. The y-intercept of the line <math>y = x + a</math> is <math>(0,\,a)</math>. We want to find the values (if any) such that <math>a=a^2</math>. Solving for <math>a</math>, the only values that satisfy this are <math>0</math> and <math>1</math>, so the answer is <math>\boxed{\mathrm{(C)}\ 2}</math> | |
== See also == | == See also == | ||
| − | + | {{AMC12 box|year=2005|ab=B|num-b=7|num-a=9}} | |
| + | {{MAA Notice}} | ||
Latest revision as of 17:23, 9 September 2020
Problem
For how many values of 
is it true that the line 
passes through the
vertex of the parabola 
?
Solution
We see that the vertex of the quadratic function is 
. The y-intercept of the line is 
. We want to find the values (if any) such that 
. Solving for , the only values that satisfy this are 
and 
, so the answer is 
See also
| 2005 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 7 |
Followed by Problem 9 |
| 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. 
