1978 AMC 12 Problem 11

Attempt Problem 11 of the 1978 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1978 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

11.

r

is positive and the line whose equation is

x+y=r

is tangent to the circle whose equation is

x^2+y^2=r,

then

r

equals

\frac12

1

2

\sqrt2

2\sqrt2

The circle has center at the origin and radius

\sqrt r

Set the distance from the origin to the line equal to the circle’s radius

The distance from the origin to

x+y-r=0

\frac{r}{\sqrt2}.

Tangency requires this to equal the radius

\sqrt r,

\frac r{\sqrt2}=\sqrt r.

Since

r\gt0,

squaring and dividing by

r

gives

r=2.

Therefore, the correct answer is C.

Problem 11 in other years