Q. The distinct points $A(0,\ 0),\ B(0,\ 1),\ C(1,\ 0)$ and $D(2a,\ 3a)$ are concyclic, then
A) ‘$a$’ can attain only rational values
B) ‘$a$’ is irrational
C) Cannot be concyclic for any ‘$d$’
D) None of the above
Explanation
(a)
Since, $B$ and $C$ are the ends of diameter as $\angle BAC$ is $90{^\circ}$
$\therefore$ Equation of circle is
$x(x - 1) + y(y - 1) = 0$
$\Rightarrow x^{2} + y^{2} - x - y = 0$
Now, point $D$ satisfies this equation
$\Rightarrow 4a^{2} + 9a^{2} - 5a = 0$
$\Rightarrow a(13a - 5) = 0$
$\Rightarrow a = 0,\ a = \frac{5}{13}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.