Circle and a Point

$\left(1,2\sqrt{2}\right)$ is a point on circle $x^2+y^2=9$. Which of the following is a point on the circle at $2$ units distance from $\left(1,2\sqrt{2}\right)$?

The maximum distance of any point on the circle ${\left(x-7\right)}^2+{\left(y-2\sqrt{30}\right)}^2=16$ from the Centre of the ellipse $25x^2+16y^2=400$is 

Find the greatest distance of the point $P\left(10, 7\right)$ from the circle $x^2+y^2-4x-2y-20=0$.

The equation of the radical axis of the circles $x^2+y^2+4x+6y-7=0$ and $4\left(x^2+y^2\right)+8x+12y-9=0$ is

If the points $\left(1,3\right)$ and $(2,k)$ are conjugated with respect the circle $x^2+y^2=35$, then the value of $k$ is

If the point $\left(1,4\right)$ lies inside the circle $x^2+y^2-6x-10y+p=0$ and the circle does not touch or intersect the coordinates axes, then

If $\mathrm{P}(2,8)$ is an interior point of a circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+4\mathrm{y}-\mathrm{p}=0$ which neither touches nor intersects the axes, then set for $\mathrm{p}$ is

The point $\left(\left[P+1\right], \left[P\right]\right),$ (where $\left[.\right]$ denotes the greatest integer function) lying inside the region bounded by the circle $x^2+y^2-2x-15=0$ and $x^2+y^2-2x-7=0,$ then :

If the point $\left(m,1\right)$ lies in the smaller region enclosed by $x^2+y^2-3x+1=0$ and $2x-y=2$ and $\lambda =\left[m\right],$ then the value of $\frac{1}{\lambda }$ is equal to (where $\left[\mathrm{.}\right]$ represents greatest integer function)

From the point $A\left(\mathrm{0,3}\right)$ on the circle $x^2+9x+{\left(y-3\right)}^2=0,$ a chord $AB$ is drawn and extended to a point $M$, such that $AM=2AB$ ( $B$ lies between $A$ & $M$). Then, the locus of the point $M$ is

If the minimum distance between the curves $y={\cos }^{-1}\left(\frac{\left(x^2-4x+5\right){\sin }^{-1}\left(1-x\right)}{{\cot }^{-1}\left({\cos }^{-1}\left(3-x\right)\right)}\right) \& {\left(x-8\right)}^2+{\left(y-\pi \right)}^2=1$ is equal to $\alpha$, then the value of $\frac{25}{\alpha }$ is :

If a chord of the circle $x^2+y^2=8$ makes equal intercepts of length $a$ on the coordinate axes and the range of values of $\left|a\right|$ is $(0, k),$ then find $\left[k\right],$ where [] represents the greatest integer function.

If the points $A:\left(0,a\right), B:\left(-2,0\right)$ and $C:\left(1,1\right)$ form an obtuse angle triangle (obtuse angled at angle $A$), then sum of all the possible integral values of $a$ is

If $M$ and $m$ are the maximum and minimum values of $\frac{y}{x}$ for pair of real numbers $\left(x,y\right)$ which satisfy the equation ${\left(x-3\right)}^2+{\left(y-3\right)}^2=6$, then the value of $\frac{1}{M}+\frac{1}{m}$ is

The point at which the line segment joining $A\left(\mathrm{1,1}\right)$ and $B\left(\mathrm{5,5}\right)$ subtends an obtuse angle is

From the point $A\left(\mathrm{0,3}\right)$ on the circle $x^2+9x+{\left(y-3\right)}^2=0,$ a chord $AB$ is drawn and extended to a point $M$ such that $AM=2AB$ ( $B$ lies between $A$ & $M$). The locus of the point $M$ is

Let $\left(a,\beta \right)$ be an ordered pair of real numbers satisfying the equation $x^2-4x+y^2+3=0.$ If the maximum and minimum values of $\sqrt{a^2+{\beta }^2}$ are $l$ and $s$ respectively, then the value of $2\left(\frac{l-s}{l+s}\right)$ is equal to

Let the points $A:\left(0,a\right), B:\left(-\mathrm{2,0}\right)$ and $C:\left(1,1\right)$ form an obtuse angled triangle (obtuse angled at angle $A$), then the complete set of values of $a$ is

The value of $a$ such that the power of the point $\left(1, 6\right)$ with respect to the circle $x^2+y^2+4x-6y-a=0$ is $-16$ is

The sum of the minimum and maximum distances of the point $(4, -3)$ to the circle $x^2+y^2+4x-10y-7=0$ is