Problem:
The graphs of and intersect at points and . Find .
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The first graph is an inverted '-shaped' right angle with vertex at and the second is a -shaped right angle with vertex at . Thus , and are consecutive vertices of a rectangle. The diagonals of this rectangle meet at their common midpoint, so the -coordinate of this mid-point is . Thus .
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Use the given information to obtain the equations , and . Subtract the third from the first to eliminate and subtract the fourth from the second to eliminate . The two resulting equations and can be solved for and . To solve the former, first consider all , for which the equation reduces to , which has no solutions. Then consider all in the interval , for which the equation reduces to , which yields . Finally, consider all , for which the equation reduces to , which has no solutions. The other equation can be solved similarly to show that . Thus .