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Q.
Four distinct points $(2k, 3k), (1, 0), (0, 1)$ and $(0, 0)$ lie on a circle for
Conic Sections
Solution:
The equation of the circle through $\left(1, 0\right)$, $\left(0,1\right)$
and $\left(0, 0\right)$ is $x^{2} + y^{2} - x - y = 0$
It passes through $\left(2k, 3k\right)$.
So, $4 k^{2} + 9k^{2} - 2k - 3k = 0$
or $13 k^{2} - 5k = 0$
$\Rightarrow \, k\left(13k - 5\right) = 0$
$\Rightarrow \, k = 0$
or $k=\frac{5}{13}\cdot$
But, $k \ne0$ [$\because$ all the four points are distinct]
$\therefore \, k=\frac{5}{13}\cdot$