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

Find the equation of the circle with center $(2, 1)$ and touching the $X$-axis.

If three distinct points $A, B, C$ are given in the $2$-dimensional coordinate plane such that the ratio of the distance from $(1, 0)$ to the distance from $(-1, 0)$ is equal to $\frac{1}{2}$,then the circumcentre of the triangle $ABC$ is at the point

Let in an equilateral $\Delta ABC,$ $A(-1 + a \cos \theta, 2 + a \sin \theta),$ $B(-1 + a \cos \alpha, 2 - a \sin \alpha),$ and $C(-1 + a \sin \beta, 2 + a \cos \beta).$ If the length of the median through vertex $A$ is $2b,$ then the equation of the circumcircle of triangle $ABC$ is (where $a$ is a constant) -

Let a circle passing through $(2,0)$ have its centre at the point $(h, k)$. Let $(x_c, y_c)$ be the point of intersection of the lines $3x + 5y = 1$ and $(2+c)x + 5c^2y = 1$. If $h = \lim_{c \to 1} x_c$ and $k = \lim_{c \to 1} y_c$,then the equation of the circle is:

The centre of the circle given by the parametric equations $x = -1 + 2\cos \theta$ and $y = 3 + 2\sin \theta$ is: