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Difference between revisions of "1978 AHSME Problems/Problem 11"

(Created page with "The circle <math>x^2 + y^2 = r</math> has center <math>(0,0)</math> and radius <math>\sqrt{r}</math>. Therefore, if the line <math>x + y = r</math> is tangent to the circle <m...")
(No difference)

Revision as of 09:28, 24 March 2015

The circle $x^2 + y^2 = r$ has center $(0,0)$ and radius $\sqrt{r}$. Therefore, if the line $x + y = r$ is tangent to the circle $x^2 + y^2 = r$, then the distance between $(0,0)$ and the line $x + y = r$ is $\sqrt{r}$.

The distance between $(0,0)$ and the line $x + y = r$ is \[\frac{|0 + 0 - r|}{\sqrt{1^2 + 1^2}} = \frac{r}{\sqrt{2}}.\] Hence, \[\frac{r}{\sqrt{2}} = \sqrt{r}.\] Then $r = \sqrt{r} \cdot \sqrt{2}$, so $\sqrt{r} = \sqrt{2}$, which means $r = \boxed{2}$ or (B), $2$.

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