2006 AMC 10A Problems/Problem 8
Problem
A parabola with equation passes through the points and . What is ?
Solution 1
Substitute the points and into the given equation for .
Then we get a system of two equations:
Subtracting the first equation from the second we have:
Then using in the first equation:
.
Solution 1.1
Alternatively, notice that since the equation is that of a conic parabola, the vertex is likely . Thus, the form of the equation of the parabola is . Expanding this out, we find that .
Solution 2
The points given have the same -value, so the vertex lies on the line .
The -coordinate of the vertex is also equal to , so set this equal to and solve for , given that :
Now the equation is of the form . Now plug in the point and solve for :
.
Solution 3
Substituting y into the two equations, we get:
Which can be written as:
and are the solutions to the quadratic. Thus:
.
See also
2006 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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All AMC 10 Problems and Solutions |
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