Let a circle passing through $\left(2 , 0\right)$ have its centre at
the point (h, k). Let $\left(x_{{}^{c }}, y_c\right)$ be the point of
intersection of the lines $3x + 5y = 1$ and $(2 + c) x + 5c^{2}y = 1.$

 If $h=\underset{c\to l}{lim{x}_c} and k= \underset{c\to l}{lim{y}_{c }},$

then the equation of the circle is :

Consider the circle $\mathrm{C} : {\mathrm{x}}^2+ {\mathrm{y}}^2= 4$ and the parabola

$\mathrm{P} : {\mathrm{y}}^2 = 8\mathrm{x}$. If the set of all values of $\alpha$, for which
three chords of the circle C on three distinct lines
passing through the point $(\mathrm{\alpha },0)$ are bisected by the
parabola P is the interval (p, q), then $(2\mathrm{q} – \mathrm{p}{)}^2$ is equal to _____.

Let the equation of the circle, which touches x-axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y-axis be ${\mathrm{x}}^2 + {\mathrm{y}}^2-\mathrm{\alpha x} + \mathrm{\beta y} + \mathrm{\gamma } = 0.$ If the circle lies below x-axis, then the ordered pair $(2\mathrm{a}, {\mathrm{b}}^2)$ is equal to :