$ax - y + c = 0$ is the equation of the common tangent to the parabola $y^2 = 8\sqrt{5}x$ and the circle $x^2 + y^2 = 1$. If this tangent makes an acute angle with the positive $X$-axis,then $a^2c^2 =$

The shortest distance between the curves $y^2=8x$ and $x^2+y^2+12y+35=0$ is:

The vertex of the parabola $(y - 1)^2 = 8(x - 1)$ is at the centre of a circle and the parabola cuts that circle at the ends of its latus rectum. Then the equation of that circle is

For a circle of diameter $R$,touching the circle $x^2 + y^2 - 4y = 0$ and passing through the point $(4, 5)$,which of the following is correct?

The focal chord to $y^2 = 16x$ is tangent to $(x - 6)^2 + y^2 = 2$. Then,the possible values of the slope of this chord are:

Let the tangent and normal at the point $(3 \sqrt{3}, 1)$ on the ellipse $\frac{x^2}{36} + \frac{y^2}{4} = 1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $AB$ as a diameter and the line $x = 2 \sqrt{5}$ intersect $C$ at the points $P$ and $Q$. If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha, \beta)$,then $\alpha^2 - \beta^2$ is equal to