IIT-JEE 1985Question 1
State TRUE or FALSE: No tangent can be drawn from the point $(5/2,\,1)$ to the circumcircle of the triangle with vertices $(1,\sqrt3)$, $(1,-\sqrt3)$, $(3,\sqrt3)$.
Circle
Solution
Right angle at $(1,\sqrt3)$, so the circumcircle has diameter from $(1,-\sqrt3)$ to $(3,\sqrt3)$: centre $(2,0)$, $r^2=4$. Distance$^2$ of $(5/2,1)$ from centre is $\frac54<4$, so the point is inside; no tangent. TRUE.