2022 AMC 12B Problems/Problem 14
The graph of intersects the -axis at points and and the -axis at point . What is ?
Solution 1 (Dot Product)
First, find , , and . Create vectors and These can be reduced to and , respectively. Then, we can use the dot product to calculate the cosine of the angle (where ) between them:
Note that intersects the -axis at points and . Without loss of generality, let these points be and respectively. Also, the graph intersects the -axis at point .
Let point . It follows that and are right triangles.
Like above, we set to , to , and to , then finding via the Pythagorean Theorem that and . Using the Law of Cosines, we see that Then, we use the identity to get
We can reflect the figure, but still have the same angle. This problem is the same as having points , , and , where we're solving for angle FED. We can use the formula for to solve now where is the -axis to angle and is the -axis to angle . and . Plugging these values into the formula, we get which is
~mathboy100 (minor LaTeX edits)
We use the formula
Note that has side-lengths and from Pythagorean theorem, with the area being
We equate the areas together to get: from which
From Pythagorean Identity,
Then we use , to obtain
Solution 6 (Complex Numbers)
From , we may assume, without loss of generality, that -intercepts of the given parabola are and . And, point has coordinates . Consider complex numbers and whose arguments are and , respectively. Notice that is the argument of the product which is Hence
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