Question

Four distinct points $(2k,3k),(1,0),(0,1)$ and $\left(0.0\right)$ lie on circle for

• A. All integral values of $k$
• B. $0<k<1$
• C. $k<0$
• D. For one value of $k$

Answer 1

The correct option is D For one value of $k$

The equation of the circle passing through $(1,0),(0,1)$ and $(0,0)$ is

$(x-1)(x-0)+(y-0)(y-1)=0\Rightarrow x^2+y^2-x-y=0$

$(\because AB$ as diameter$)$

which passes through $(2k,3k)$

then $4k^2+9k^2-2k-3k=0\Rightarrow 13k^2-5k=0$

As $k\ne 0,\therefore k=\frac{5}{13}$

$(k\ne 0$ as the four points are distinct$)$