smc

Question

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:

Accepted Answer

By definition of mean proportional, t = mn. Since m+n=10, t = 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 two.