If $(\alpha, \beta)$ is a point on the circle whose centre is on the $x$-axis and which touches the line $x + y = 0$ at $(2, -2)$,then the greatest value of $\alpha$ is

An equilateral triangle $OAB$ is inscribed in the parabola $y^{2}=4x$ with the vertex $O$ at the origin. Then the minimum distance of the circle having $AB$ as a diameter from the origin is:

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$ and the circle $x^{2}+y^{2}=3$ at their point of intersection in the first quadrant. Then $\tan \theta$ is equal to:

The equation of the common tangent touching the circle $(x - 3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the $X$-axis is

Let $a$ and $b$ be two non-zero real numbers. The equation $(ax^2 + by^2 + c)(x^2 - 5xy + 6y^2) = 0$ represents:

Consider the equations of the circles:
$S_1 : x^2 + y^2 + 24x - 10y + a = 0$
$S_2 : x^2 + y^2 = 36$
Which of the following statements is not correct?