Question

Four distinct points $(2K,3K),(1,0),(0,1)$ and $(0,0)$ lie on a circle when

• A. all values of $K$ are integral
• B. $0<K<1$
• C. $K<0$
• D. For one values of $K$

Answer 1

The correct option is D For one values of $K$

Let A = $(0,0)$ , B$(0,1)$ , C$(2k,3k)$ , D$(1,0)$

As we can see

$AD\perp AB$

$\angle A=90$

$\angle C=90$

$⟹m_{BC}\times m_{DC}=-1$

$\frac{3k-1}{2k}\times 3k2k-1=-1$

$⟹k(9k-3)=-(4k-2)k$

$k=0,9k+4k=2+3⟹13k=5⟹k=\frac{5}{13}$

But if $k=0$, C will be $(0,0)$ which is A

$k=\frac{5}{13}$ only one point.