Coordinate Geometry
Circles
MMTS_Full_Test_15
Grade 12
Question:
If a circle of radius $r$ passes through origin and meets axes at $A$ and $B$ such that $(3,4)$ lies on $\overline{AB}$, then $r$ is
$\dfrac{25}{8}$
$\dfrac{25}{6}$
$\dfrac{25}{3}$
$\dfrac{25}{4}$
Step-by-Step Solution
Key Concept: Circle through origin with diameter endpoints on axes: $A=(a,0)$, $B=(0,b)$; equation $x(x-a)+y(y-b)=0$; radius$=\sqrt{a^2+b^2}/2$
$3/a+4/b=1$. $r^2=(a^2+b^2)/4$. Minimize: by AM-GM with constraint, $r=25/8$.
Correct Answer: 1