Please Pass Me the Sun Tangent Lotion

Say the tangent point is $P(u,v)$ . The radius $OP$ has slope $\frac{v}{u}$ ; since the product of slopes of perpendicular lines is $-1$ , the tangent has slope $\frac{-u}{v}$ . We're told this is a positive prime; call it $p$ .

Say the $y$ -intercept has coordinates $(0,q)$ . Then $\frac{v-q}{u-0}=p$ so that $q=v-pu$ . Now, from the slope equation, $u=-pv$ ; substituting in gives $q=v\left(1+p^2\right)$

Since $q$ is a positive prime, and $1+p^2$ is greater than $1$ , we must have $v=1$ . If $p$ were odd, $1+p^2$ would be even (and greater than $2$ ), so not a prime; hence $p$ is even. Since the only even prime is $2$ , we get $p=2$ , $q=5$ , $v=1$ and $\boxed{u=-2}$ .