The equation of the pair of tangents drawn from the point $(0, 1)$ to the circle $x^2 + y^2 - 2x + 4y = 0$ is . . . . . .

Tangents are drawn from the point $(-1, -4)$ to the circle $x^2 + y^2 - 2x + 4y + 1 = 0$. The length of the corresponding chord of contact is:

Tangents are drawn from the point $(4, 3)$ to the circle $x^2 + y^2 = 9$. The area of the triangle formed by them and the line joining their points of contact is

Consider two circles $C_1: x^2+y^2=25$ and $C_2: (x-\alpha)^2+y^2=16$,where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of the common chord of $C_1$ and $C_2$ is $\beta$,then the value of $(\alpha \beta)^2$ equals:

Chords of the curve $4x^2 + y^2 - x + 4y = 0$ which subtend a right angle at the origin pass through a fixed point whose coordinates are:

If the line $x+y+1=0$ intersects the circle $x^2+y^2+x+3y=0$ at two points $A$ and $B$,then the centre of the circle which passes through the points $A, B$ and the point of intersection of the tangents drawn at $A$ and $B$ to the given circle is