2015 AMC 12A Problems/Problem 12
Problem
The parabolas and
intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area
. What is
?
Solution
Clearly, the parabolas must intersect the x-axis at the same two points. Their distance multiplied by (the distance between the y-intercepts), all divided by 2 is equal to 12, the area of the kite (half the product of the diagonals). That distance is thus 4, and so the x-intercepts are
Then
, and
Then
.
Alternative Solution
The parabolas must intersect the x-axis at the same two points for the kite to form. We find the x values at which they intersect by equating them and solving for x as shown below. and
or
. The x-values of the y-intercepts is 0, so we plug in zero in each of them and get
and
. The area of a kite is
. The
is
. The
is
. Solving for the area
, therefore
.
See Also
2015 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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 |