1960 AHSME Problems/Problem 35

Problem 35

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$ . It is found that $t_1$ , the length of the tangent, is the mean proportional between $m$ and $n$ . If $m$ and $t$ are integers, then $t$ may have the following number of values:

$\textbf{(A)}\ \text{zero}\qquad \textbf{(B)}\ \text{one}\qquad \textbf{(C)}\ \text{two}\qquad \textbf{(D)}\ \text{three}\qquad \textbf{(E)}\ \text{infinitely many}$

Solution

By definition of mean proportional, $t = \sqrt{mn}$ . Since $m+n=10$ , $t = \sqrt{m(10-m)}$ .

With trial and error, note that when $m=9$ , $t=3$ and when $m=2$ , $t=4$ . These values work since another tangent line can be drawn from $P$ , and the angle between the tangent and secant can decrease to match the values of $m$ and $n$ . Thus, the answer is $\boxed{\textbf{(C)}}$ .

1960 AHSC (Problems • Answer Key • Resources)
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1960 AHSME Problems/Problem 35

Problem 35

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