1957 AHSME Problems/Problem 23

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 $\boxed{\textbf{(C) }\sqrt{2}}$

1957 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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1957 AHSME Problems/Problem 23

Solution

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