If $C(\alpha, \beta)$ with $\alpha < 0$ is the centre of the circle that touches the $Y$-axis at $(0, 3)$ and makes an intercept of length $2$ units on the positive $X$-axis,then $(\alpha, \beta) =$

If $(1, a)$ and $(b, 2)$ are conjugate points with respect to the circle $x^2+y^2=25$,then $4a+2b=$

If a line passing through the point $\left( -\sqrt{8}, \sqrt{8} \right)$ and making an angle of $135^{\circ}$ with the $x$-axis intersects the circle $x = 5 \cos \theta, y = 5 \sin \theta$ at points $A$ and $B$,then find the length of the chord $AB$.

For each natural number $k$,let $C_k$ denote the circle with radius $k$ centimeters and center at the origin. On the circle $C_k$,a particle moves $k$ centimeters in the counter-clockwise direction. After completing its motion on $C_k$,the particle moves to $C_{k+1}$ in the radial direction. The motion of the particle continues in this manner. The particle starts at $(1, 0)$. If the particle crosses the positive direction of the $x$-axis for the first time on the circle $C_n$,then $n$ is equal to

The length of the chord intercepted by the circle $x^2 + y^2 = r^2$ on the line $\frac{x}{a} + \frac{y}{b} = 1$ is

The coordinates of the mid-point of the chord cut off by the line $2x - 5y + 18 = 0$ by the circle $x^{2} + y^{2} - 6x + 2y - 54 = 0$ are: