Equation of a circle

Let $x^2+y^2+Ax+By+C=0$ be a circle passing through $(0,6)$ and touching the parabola $y=x^2$ at $(2,4)$. Then $A+C$ is equal to ______.

Let $S=\{z\in\mathbb C:|z-3|\le1$ and $z(4+3i)+\bar z(4-3i)\le24\}$. If $\alpha+i\beta$ is the point in $S$ which is closest to $4i$, then $25(\alpha+\beta)$ is equal to ______.

Let a circle $C:(x-h)^2+(y-k)^2=r^2$, $k>0$, touch the $x$-axis at $(1,0)$. If the line $x+y=0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is 2, then the value of $h+k+r$ is equal to ______.

Let a circle C touch the lines $L_1:4x-3y+K_1=0$ and $L_2:4x-3y+K_2=0$, $K_1,K_2\in\mathbb R$. If a line passing through the centre of the circle C intersects $L_1$ at $(-1,2)$ and $L_2$ at $(3,-6)$, then the equation of the circle C is :

Let the abscissae of the two points P and Q be the roots of $2x^2-rx+p=0$ and the ordinates of P and Q be the roots of $x^2-sx-q=0$. If the equation of the circle described on PQ as diameter is $2\left(x^2+y^2\right)-11x-14y-22=0$, then $2r+s-2q+p$ is equal to ______.

A circle touches both the $y$-axis and the line $x+y=0$. Then the locus of its center is:

Let C be a circle passing through the points A$(2,-1)$ and B$(3,4)$. The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle $(x-5)^2+(y-1)^2=\dfrac{13}2$, then $r^2$ is equal to :

Let $\dfrac{dy}{dx}=\dfrac{ax-by+a}{bx+cy+a}$, where $a,b,c$ are constants, represent a circle passing through the point $(2,5)$. Then the shortest distance of the point $(11,6)$ from this circle is :

A rectangle R with end points of one of its sides as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of R is ______.

A circle of radius 2 unit passes through the vertex and the focus of the parabola $y^2=2x$ and touches the parabola $y=\left(x-\dfrac14\right)^2+\alpha$, where $\alpha>0$. Then $(4\alpha-8)^2$ is equal to ______.

The set of values of $k$, for which the circle $C:4x^2+4y^2-12x+8y+k=0$ lies inside the fourth quadrant and the point $\left(1,-\dfrac13\right)$ lies on or inside the circle C, is

If one of the diameters of the circle $x^2+y^2-2\sqrt2x-6\sqrt2y+14=0$ is a chord of the circle $\left(x-2\sqrt2\right)^2+\left(y-2\sqrt2\right)^2=r^2$, then the value of $r^2$ is equal to ______.

Let a triangle ABC be inscribed in the circle $x^2-\sqrt2(x+y)+y^2=0$ such that $\angle BAC=\dfrac\pi2$. If the length of side AB is $\sqrt2$, then the area of the $\Delta ABC$ is equal to :

Let $\dfrac{dy}{dx}=\dfrac{ax-by+a}{bx+cy+a}$, $a,b,c\in\mathbf R$, represents a circle with center $(\alpha,\beta)$. Then, $\alpha+2\beta$ is equal to :

For $n\in\mathbf N$, let $S_n=\left\{z\in\mathbf C:|z-3+2i|=\dfrac n4\right\}$ and $T_n=\left\{z\in\mathbf C:|z-2+3i|=\dfrac1n\right\}$. Then the number of elements in the set $\{n\in\mathbf N:S_n\cap T_n=\phi\}$ is :

Let the equation of two diameters of a circle $x^2+y^2-2x+2fy+1=0$ be $2px-y=1$ and $2x+py=4p$. Then the slope $m\in(0,\infty)$ of the tangent to the hyperbola $3x^2-y^2=3$ passing through the centre of the circle is equal to ______.

A point P moves so that the sum of squares of its distances from the points (1, 2) and (−2, 1) is 14. Let $f(x,y)=0$ be the locus of P, which intersects the $x$-axis at the points A, B and the $y$-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to :

Let P and Q be any points on the curves $(x-1)^2+(y+1)^2=1$ and $y=x^2$, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval

Let the abscissae of the two points P and Q on a circle be the roots of $x^2-4x-6=0$ and the ordinates of P and Q be the roots of $y^2+2y-7=0$. If PQ is a diameter of the circle $x^2+y^2+2ax+2by+c=0$, then the value of $(a+b-c)$ is ______.

The area of the smaller region enclosed by the curves $y^2=8x+4$ and $x^2+y^2+4\sqrt3x-4=0$ is equal to