The distance from the centre of the circle $x^2 + y^2 = 2x$ to the straight line passing through the points of intersection of the two circles $x^2 + y^2 + 5x - 8y + 1 = 0$ and $x^2 + y^2 - 3x + 7y - 25 = 0$ 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:

The chord of contact of the point $(3, 2)$ with respect to the circle $x^2 + y^2 = 25$ meets the coordinate axes at $A$ and $B$. The circumcentre of triangle $OAB$ is

The intercept on the line $y = x$ by the circle ${x^2} + {y^2} - 2x = 0$ is $AB$. The equation of the circle having $AB$ as a diameter is

Let the tangents at two points $A$ and $B$ on the circle $x^{2} + y^{2} - 4x + 3 = 0$ meet at the origin $O(0,0)$. Then the area of the triangle $OAB$ is:

$C_1$ is the circle with centre at $O(0,0)$ and radius $4$,$C_2$ is a variable circle with centre at $(\alpha, \beta)$ and radius $5$. If the common chord of $C_1$ and $C_2$ has slope $\frac{3}{4}$ and is of maximum length,then one of the possible values of $\alpha+\beta$ is