1957 AHSME Problems/Problem 23

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The graph of $x^2 + y = 10$ and the graph of $x + y = 10$ meet in two points. The distance between these two points is:

$\textbf{(A)}\ \text{less than 1} \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ \sqrt{2}\qquad \textbf{(D)}\ 2\qquad\textbf{(E)}\ \text{more than 2}$

Solution

We can merge the two equations to create $x^2+y=x+y$. Using either the quadratic equation or factoring, we get two solutions with $x$-coordinates $0$ and $1$.

Plugging this into either of the original equations, we get $(0,10)$ and $(1,9)$. The distance between those two points is $\boxed{\textbf{(C) }\sqrt{2}}$.

See Also

1957 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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