Difference between revisions of "2005 AMC 12B Problems/Problem 8"
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\mathrm{(D)}\ 10 \qquad | \mathrm{(D)}\ 10 \qquad | ||
\mathrm{(E)}\ \text{infinitely many} | \mathrm{(E)}\ \text{infinitely many} | ||
+ | \mathrm{(F)}\ \infty \qquad | ||
</math> | </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> | 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> |
Revision as of 18:23, 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.