Equation of Circle

Tangents drawn from the point $P(1, 8)$ to the circle 

$x^2+y^2-6x-4y-11=0$

touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $PAB$ is

Extremities of a diagonal of a rectangle are $\left(0, 0\right)$ and $\left(4, 3\right).$ Find the equation of the tangents to the circumcircle of a rectangle which are parallel to this diagonal.

The circle $C: x^2+y^2+kx+\left(1+k\right)y-\left(k+1\right)=0$ passes through two fixed points for every real number $K$. The minimum value of the radius of circle $C$ is

If $A$ and $B$ are fixed points in the plane such that $\frac{PA}{PB}=k$  (constant) for all $P$ on a given circle, then the value of $k$ cannot be equal to:

Two circles in the first quadrant of radii $r_1$ and $r_2$ touch the coordinate axes. Each of them cuts off an intercept of $2$ units with the line $x+y=2$. Then ${r_1}^2+{r_2}^2-r_1{r}_2$ is equal to ____.

Let $A$ be the point $\left(1,2\right)$ and $B$ be any point on the curve $x^2+y^2=16$. If the centre of the locus of the point $P$, which divides the line segment $A B$ in the ratio $3:2$ is the point $C\left(\alpha ,\beta \right)$, then the length of the line segment $AC$ is

A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio $2:3$, is a circle of radius

A circle passing through the point $P\left(\alpha ,\beta \right)$ in the first quadrant touches the two coordinate axes at the points $A$ and $B.$ The point $P$ is above the line $AB.$ The point $Q$ on the line segment $AB$ is the foot of perpendicular from $P$ on $AB.$ If $PQ$ is equal to $11$ units, then the value of $\alpha \beta$ is $\_\_\_\_\_\_\_$

Let $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^2+y^2-6x+4y+8=0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $OPQ$ passes through the point $\left(\mathrm{\alpha },\frac{1}{2}\right)$, then a value of $\alpha$ is

Two circles having radius $r_1 \& r_2$ touch both the coordinate axes, if line $x+y=2$ makes the intercept $2$ on both the circles then the value of ${r_1}^2+{r_2}^2-r_1{r}_2$ will be 

From $O\left(0,0\right)$ two tangents $OA$ and $OB$ are drawn to a circle $x^2+y^2-6x+4y+8=0$ then the equation of circumircle of $∆OAB$ is

Four distinct points $\left(a,\frac{1}{a}\right), \left(b,\frac{1}{b}\right), \left(c,\frac{1}{c}\right)$ and $\left(d,\frac{1}{d}\right)$ lie on a circle where $a,b,c,d\ne 0$, then the value of $abcd=$

The equation of circle with radius $3$ and centre as the point of intersection of the lines $2x+3y=5$, $2x-y=1$ is

Centre and radius of the circle with segment of the line $x+y=1$ cut off by coordinate axes as diameter is

If the circle $x^2+y^2+2x-4y-k=0$ is mid-way between the circles $x^2+y^2+2x-4y-4=0$ and $x^2+y^2+2x-4y-20=0$, then $k=$

If two circles $x^2+y^2+2gx+c=0$ and $x^2+y^2-2fy-c=0$ have equal radius, then locus of $\left(g,f\right)$ is

The equation of the circle with $x+y-5=0$ and $x-2y-2=0$ as diameters and having $3x+4y-1=0$ as tangent is

The circle touches the $Y-\mathrm{axis}$ at the point $\left(0,4\right)$ and cuts the $X-\mathrm{axis}$ in a chord of length $6$ units. The radius of the circle is

If the radius of circle which touches $x$-axis at origin and the curve $y=\frac{1}{x}$ is '$r$', then the value of $3\sqrt{3}{r}^2$ is