General Equation of a Circle

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

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 $abcd=$

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

If the circle $x^2+y^2+2gx+2fy+c=0$ touches $x-$axis then

A rectangle $ABCD$ is inscribed in the circle $x^2+y^2+3x+12y+2=0$. If the co-ordinates of $A$ and $B$ are $\left(3,-2\right) \& \left(-2,0\right)$, then the other two vertices of the rectangle are:

Let a circle represented as $f\left(x,y\right)=0$. If $f\left(k,0\right)$ has repeated roots as $k=1,1$ and $f\left(0,k\right)$ has roots as $k=\frac{1}{2}, 2$ and centre of the circle is given by $\left(m,n\right)$, then $4\left(m+n\right)=$

A circle passes through the points $\left(1,2\right),\left(3,4\right)$. If its centre lies on the line $x-y+3=0$, then its radius is equal to

A circle has its centre in the first quadrant and passes through $\left(2,3\right)$. If this circle makes intercepts of length $3$ and $4$ respectively on $x=2$ and $y=3$, its equation is

If the radius of the circle $x^2+y^2+ax+by+3=0$ is $2$, then the point $\left(a,b\right)$ lies on the circle

Let $f\left(x, y\right)=0$ be the equation of circle. If $f\left(0, \lambda \right)=0$ has equal roots $\lambda =1, 1$ and $f\left(\lambda , 0\right)=0$ has roots $\lambda =\frac{1}{5}, 5,$ then the radius of the circle is

The equation of a circle with centre at $\left(2,2\right)$ and passes through the point $\left(4,5\right)$ is

The equation of a circle with center at $\left(-2, 3\right)$ and circumference of $4\pi$ units is

The radius of a circle whose centre is $(2, 1)$ and one of the chords is a diameter of the circle $x^2+y^2-2\mathrm{x}-6\mathrm{y}+6=0,$ is $\_\_\_\_\_\_\_\_\_\_\_$ units.

The number of integer values of $k$ for which the equation $x^2+y^2+\left(k-1\right)x-ky+5=0$ represents a circle whose radius cannot exceed $3$, is

The area of the circle $x^2-2x+y^2-10y+k=0$ is $25\pi .$ The value of $k$ is equal to

On what condition the equation $a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$ may represent a circle?

Show that the locus of a point, such that the ratio of its distances from two given points is constant $(\ne 1)$, is a circle. Hence show that the circle cannot pass through the given points.

Find the values of $m$ for which the family of the lines $y=mx+2$,

(a) may interesect the circle $S=0$ in two distinct points,

(b) may touch the circle $S=0$, and

(c) does not meet the circle $S=0$ where $\mathrm{S}=x^2+y^2+3x+3y-8$

On the circle $16x^2+16y^2+48x-8y=43$, find the point $P$ nearest to the line $8x-4y+73=0$ and calculate the distance between the point $P$ and the line $8x-4y+73=0$