Question

Four distinct points (2k, 3k), (1, 0) (0, 1) and (0, 0) lie on a circle for

• A. ∀ k ∈ I
• B. k < 0
• C. 0 < k < 1
• D. For two values of k

Answer 1

The correct option is D For two values of k

General equation of circle is,
$x^2+y^2+2gx+2fy+c=0$

It passes through (0,0), (1, 0) and (0, 1); $\therefore$ c = 0
Now 2g + 1 = 0 $\Rightarrow$ g = $-\frac{1}{2}$ and 2f + 1 = 0 $\Rightarrow$ f = $\frac{-1}{2}$
Hence equation of circle is $x^2+y^2-x-y=0$
Point (2k, 3k) lies on the circle
$\therefore 4k^2+9k^2-5k=0$
$\Rightarrow 13k^2-5k=0$
$\Rightarrow k=0ork=\frac{5}{13}.$