1955 AHSME Problems/Problem 43

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Problem 43

The pairs of values of $x$ and $y$ that are the common solutions of the equations $y=(x+1)^2$ and $xy+y=1$ are:

$\textbf{(A)}\ \text{3 real pairs}\qquad\textbf{(B)}\ \text{4 real pairs}\qquad\textbf{(C)}\ \text{4 imaginary pairs}\\ \textbf{(D)}\ \text{2 real and 2 imaginary pairs}\qquad\textbf{(E)}\ \text{1 real and 2 imaginary pairs}$

Solution

Graph the two equations. $y=(x+1)^2$ produces a parabola with vertex $(-1, 0),$ while $xy+y=1$ produces a hyperbola. They intersect at the point $(0, 1).$

This makes one real solution. Only one answer choice has 1 real solution listed, so the answer is $\boxed{\textbf{(E)}}.$

See Also

1955 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 42
Followed by
Problem 44
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