Among the chords of the circle $x^2+y^2=75$, the number of chords having their mid-points on the line $x=8$ and having their slopes as integers is

The equation of the circle which touches the circle $S \equiv x^2+y^2-10 x-4 y+19=0$ at the point $(2,3)$ internally and having radius equal to half of the radius of the circle $S=0$ is

If $P\left(\frac{7}{5}, \frac{6}{5}\right)$ is the inverse point of $A(1,2)$ with respect to a circle with centre $C(2,0)$, then the radius of that circle is

If the circle $S=0$ intersect the three circle

$ \begin{aligned} & S_1 \equiv x^2+y^2+4 x-7=0 \\ & S_2 \equiv x^2+y^2+y=0 \text { and } S_3 \equiv x^2+y^2+\frac{3}{2} x+\frac{5}{2} y-\frac{9}{2}=0 \end{aligned} $

orthogonally, then radical axis of $S=0$ and $S_1=0$ is

If a tangent of the circle $x^2+y^2+2 x+2 y+1=0$ is radical axis of the circles $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$, then

If the length of the chord $2 x+3 y+k=0$ of the circle $x^2+y^2-2 x+4 y-11=0$ is $2 \sqrt{3}$, then the sum of all possible values of $k$ is

The power of a point $(2,-1)$ with respect to a circle $C$ of radius 4 is 9 . The centre of the circle $C$ lies on the lines $x+y=0$ and in the 2nd quadrant. If ( $\alpha, \beta$ ) is the centre of the circle $C$ then $\beta-\alpha=$

The angle between the tangents drawn from the point $P(k, 6 k)$ to the circle $x^2+y^2+6 x-6 y+2=0$ is $2 \tan ^{-1}\left(\frac{4}{3}\right)$. If the coordinates of $P$ are integers, then $k=$

The tangents drawn from a point $(2,-1)$ touch the circle $x^2+y^2+4 x-2 y+1=0$ at the points $A$ and $B$. If $C$ is the centre of the circle, then the area (in sq. units) of the $\triangle A B C$ is

If $\theta$ is the angle between the circles $x^2+y^2-4 x+2 y-4=0$ and $x^2+y^2-2 x+4 y-11=0$ then $\sin \theta=$

If the line $x+y=2$ cuts the circle $x^2+y^2+2 x-4 y+4=0$ at two points $A$ and $B$, then the radius of the circle passing through $A, B$ and orthogonal to $x^2+y^2-2 x-4 y-4=0$ is

If $(3,-2)$ is the centre of the circle $S \equiv x^2+y^2+2 g x+2 f y-23=0$ and $A$ is a point on the circle $S=0$ such that its distance from a point $P(-1,-5)$ is least, then $A=$

Two circles which touch both the coordinate axes intersect at the points $A$ and $B$. If $A=(1,2)$, then $A B=$

The lines $4 x-3 y+2=0$ intersects the circle $x^2+y^2-2 x+6 y+c=0$ at two points $A, B$ and $A B=8$. If $(1, k)$ is a point on the given circle and $k>0$, then $k=$

If $2 x-3 y+5=0$ and $4 x-5 y+7=0$ are the equations of the normals drawn to a circle and $(2,5)$ is a point on the given circle, then the radius of the circle is

If $(\alpha, \beta)$ is the centre of the circle which passes through the point $(1,-1)$ and cuts the circles

$ x^2+y^2+2 x-3 y-5=0, x^2+y^2-3 x+2 y+1=0 $

orthogonally, then $\alpha-5 \beta=$

The centre of the circle touching the circles $x^2+y^2-4 x-6 y-12=0$

$x^2+y^2+6 x+18 y+26=0$ at their point of contact and passing through the point $(1,-1)$ is

The equation of the locus of a point, which is at a distance of 5 units from a fixed point $(1,4)$ and also from a fixed line $2 x+3 y-1=0$ is

If the equation of the circumcircle of the triangle formed by the lines $L_1 \equiv x+y=0$,

$L_2 \equiv 2 x+y-1=0, L_3 \equiv x-3 y+2=0$ is $\lambda_1 L_1 L_2+\lambda_2 L_2 L_3+\lambda_3 L_3 L_1=0$, then $\frac{7 \lambda_1}{\lambda_2}+\frac{\lambda_3}{\lambda_1}=$

A circle $C$ touches $X$-axis and makes an intercept of length 2 units on $Y$-axis. If the centre of this circle lies on the line $y=x+1$, then a circle passing through the centre of the circle $C$ is

If $m_1, m_2$ are the slopes of the tangents drawn through the point $(-1,-2)$ to the circle $(x-3)^2+(y-4)^2=4$, then $\sqrt{3}\left|m_1-m_2\right|=$

A line meets the circle $x^2+y^2-4 x-4 y-8=0$ in two points $A$ and $B$. If $P(2,-2)$ is a point on the circle such that $P A=P B=2$, then the equation of the line $A B$ is

If the centre $(\alpha, \beta)$ of a circle cutting the circles $x^2+y^2-2 y-3=0$ and $x^2+y^2+4 x+3=0$ orthogonally lies on the line $2 x-3 y+4=0$, then $2 \alpha+\beta=$

The radius of a circle $C_1$ is thrice the radius of another circle $C_2$ and the centres of $C_1$ and $C_2$ are $(1,2)$ and $(3,-2)$ respectively. If they cut each other orthogonally and the radius of the circle $C_1$ is $3 r$, then the equation of the circle with $r$ as radius and $(1,-2)$ as centre is

The equation of the circle whose radius is 3 and which touches the circle $x^2+y^2-4 x-6 y-12=0$ internally at $(-1,-1)$ is

Suppose $C_1$ and $C_2$ are two circles having no common points, then

The locus of the centre of the circle touching the $X$-axis and passing through the point $(-1,1)$ is

The centres of all circles passing through the points of intersection of the circles $x^2+y^2+2 x-2 y+1=0$ and $x^2+y^2-2 x+2 y-2=0$ and having radius $\sqrt{14}$ lie on the curve

$A$ circle $S$ given by $x^2+y^2-14 x+6 y+33=0$ cuts the $X$-axis at $A$ and $B(O B>O A)$. $C$ is mid-point of $A B . L$ is a line through $C$ and having slope ( -1 ). If $L$ is the diameter of a circle $S^{\prime}$ and also the radical axis of the circles $S$ and $S^{\prime}$, then the equation of the circle $S^{\prime}$ is

If the equation of the circle passing through the points $(-1,0),(-1,1),(1,1)$ is $a x^2+a y^2+2 g x+2 f y-2=0$, then $a=$

For the circle $x-2=5 \cos \theta, y+1=5 \sin \theta$, where $\theta$ is the perimeter, the line $x=1+\frac{r}{2}, y=-2+\frac{\sqrt{3}}{2} r$ where $r$ is the perimeter, is a

If $x-2 y=0$ is a tangent drawn at a point $P$ on the circle $x^2+y^2-6 x+2 y+c=0$, then the distance of the point $(6,3)$ from $P$ is

If the angle between the circles $x^2+y^2-2 x+k y+1=0$ and $x^2+y^2-k x-2 y+1=0$ is $\cos ^{-1}\left(\frac{1}{4}\right)$ and $k<0$, then the point which lies on the radical axis of the given circle is

If the circumcenter of the triangle formed by the points $A(a, 3), B(b, 5)$ and $C(a, b)$ is $(1,1)$, then out of all the possible coordinates of $C$ the sum of the absolute values of the distinct coordinates of $C$ is

The equation of a circle passing through $(-6,3)$ and touching both the coordinates axes is

The area (in sq units) of the triangle formed by the $x$-axis, the tangent and the normal drawn to the circle $x^2+y^2=10 x$ at the point $(9,3)$ is

The number of common tangents of the circles $x^2+y^2-4=0$ and $x^2+y^2-6 x-8 y-24=0$ is

If the equation of the circle whose radius is $\sqrt{10}$ and which touches the circle $x^2+y^2+2 x+8 y-23=0$ externally at the point $(1,2)$ is $x^2+y^2+a x+b y+c=0$, then $|a+b+c|=$

If a circle ' $S$ ' passing through the origin and having its centre on the line $x-y=0$ cuts the circle $x^2+y^2-4 x-6 y+10=0$ orthogonally, then the diameter of ' $S$ ' is

The equation of the circle passing through the points of intersection of the circles $x^2+y^2+6 x+4 y-12=0$, $x^2+y^2-4 x-6 y-12=0$ and having radius $\sqrt{13}$ is

If a point $P$ moves so that the distance from $(0,2)$ to $P$ is $\frac{1}{\sqrt{2}}$ times the distance of $P$ from $(-1,0)$, then the locus of the point $P$ is

If the parametric equations of the circle passing through the points $(3,4),(3,2)$ and $(1,4)$ is $x=a+r \cos \theta, y=b+r \sin \theta$, then $b^a r^a=$

A tangent $P T$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3}, 1)$. If a straight line $L$ which is perpendicular to $P T$ is a tangent to the circle $(x-3)^2+y^2=1$, then a possible equation of $L$ is

If the angle between the pair of tangents drawn to the circle $x^2+y^2-2 x+4 y+3=0$ from the point $(6,-5)$ is $\theta$, then $\cot \theta=$

If the angle between the circles $x^2+y^2-4 x-6 y+k=0$ and $x^2+y^2+8 x-4 y+11=0$ is $\frac{\pi}{2}$, then the value of $k$ is

The radius of a circle touching all the four circles $(x \pm \lambda)^2+(y \pm \lambda)^2=\lambda^2$ is

If the radical centre of the given three circles $x^2+y^2=1, x^2+y^2-2 x-3=0$ and $x^2+y^2-2 y-3=0$ is $C(\alpha, \beta)$ and $r$ is the sum of the radii of the given circles, then the circle with $C(\alpha, \beta)$ as centre and $r$ as radius is

The equation of the circle inscribed in a square formed by the lines $x+y-2=0, x+y-6=0, x-y+1=0$ and $x-y+5=0$ is

Let the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ touch the positive $X$-axis and the positive $Y$-axis. Let $(2,4)$ be a point on the circle $S=0$. If two such circles exist, then the difference of their areas is

If the equation $2 x-3 y+3=0,2 x+y+1=0$ and $6 x+4 y+1=0$ represent the sides of a triangle, then the equation of the circle passing through the vertices of this triangle is

If $T_1 T^{\prime}{ }_1$ and $T_2 T_2^{\prime}$ are the common tangents of the circles $S \equiv x^2+y^2-2 x-4 y-4=0$ and $S \equiv x^2+y^2+4 x+4=0$, where $T_1, T^{\prime}{ }_1, T_2, T^{\prime}{ }_2$ are the points of contact, then the distance between $T_1$ and $T_1^{\prime}$ is

A circle $S \equiv x^2+y^2+2 g x+2 f y+4=0$ cuts the circle $x^2+y^2-4 x-4 y-4=0$ orthogonally and makes an angle of $60^{\circ}$ with the circle $x^2+y^2+4 x+4 y+4=0$. Then, the radius of the circle $S=0$ is

If the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ cuts each of the three circles $x^2+y^2+4 x+4 y+7=0$, $x^2+y^2-4 x+4 y+7=0$ and $x^2+y^2-4 x-4 y+7=0$ orthogonally, then the equation of the tangent drawn at the point $(\sqrt{3}, 2)$ to the circle $S=0$ is

Let a chord $A B$ subtend an angle of $60^{\circ}$ at the centre $C(2,3)$ of a circle $S$. If the equation of $A B$ is $x+y+1=0$, then the equation of the circle $S$ is

Let 6,8 be the $X$ and $Y$-intercepts made by the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$, respectively. If $g x+f y+1=0$ is a line passing through the point $(1,-1)$, then the radius of the circle $S=0$ is

If $(3,1)$ and $(-2,4)$ are points on a circle $S$ whose centre lies on the line $x-y+1=0$, then the parametric equations of $S$ are

Let $S \equiv x^2+y^2-8 x+10 y+5=0$ be a circle. Let $P(1,1)$ and $Q(1,-1)$ be two points. Then, the point of intersection of the polar of $P$ with respect to $S=0$ and the chord with $Q$ as mid-point to $S=0$ is

If the angle between the circles $x^2+y^2-2 x+2 y+1=0$ and $x^2+y^2+2 x-2 y+k=0$ is $\frac{\pi}{3}$, then

Let the line $x-y+1=0$ intersect the circle $x^2+y^2+2 x+2 y+1=0$ in two points $A$ and $B$. If $A B$ is the diameter of the circle $x^2+y^2+2 g x+2 f y+c=0$, then $g+f=$

If a circle passing through $(1,-2)$ has $x-y=2$ and $2 x+3 y=14$ as its diameters, then the radius of the circle is

The equation of the circle whose diameter is the common chord of the circles $x^2+y^2+2 x+3 y+1=0$ and $x^2+y^2+4 x+3 y+2=0$ is

The pole of the straight line $9 x+y-28=0$ with respect to the circle $2 x^2+2 y^2-3 x+5 y-7=0$ is

The equation of the line perpendicular to the radical axis of two circles $x^2+y^2-5 x+6 y+12=0$, $x^2+y^2+6 x-4 y-14=0$ and passing through $(1,1)$ is

If the angle between the circles

$ x^2+y^2-2 x-4 y+c=0 \text { and } x^2+y^2-4 x-2 y+4=0 $

is $60^{\circ}$, then $c=$

The line $4 x+3 y-4=0$ divides the circumference of a circle in the ratio $1: 2$. If $C(5,3)$ is the centre of that circle, then equation of the circle is

Two sides of a square are along the lines $x=-5$ and $y=4$. The point of intersection of the diagonals is $(3,-4)$. The point of intersection of the tangents drawn to the circumcircle of the square at the two consecutive vertices lying on $x=-5$ is

If $L_1, L_2$ and $L_3$ are the chords of contact of the three points $(2,0),(1,-2)$ and $(4,4)$ respectively with respect to the circle $x^2+y^2=3$, then $L_1, L_2$ and $L_3$ are

The combined equation of the direct common tangents of the circles $x^2+y^2+2 x=0$ and $x^2+y^2-2 y-3=0$

If $(h, k)$ is the centre of the circle which passes through the origin and cuts the circles $x^2+y^2+4 x+6 y+12=0$ and $x^2+y^2+4 x-6 y+9=0$ orthogonally, then $k-2 h=$

If $(-1,-1)$ is the radical centre of the circles $x^2+y^2+2 g x-4 y+4=0, x^2+y^2+6 x+2 f y+12=0$ and $x^2+y^2+10 y+20=0$, then $g-f=$

Let the centre of the circle $S=0$ lie on the line $x+y-5=0$ and also lie in the first quadrant. If this circle touches both the lines $x-2=0$ and $y-5=0$, then the area of the circle is

The straight line $x+2 y=1$ cuts the $X$-axis at $A$ and $Y$-axis at $B, A$ circle is drawn through $A, B$ and the origin. The sum of the perpendicular distances from $A$ and $B$ on to the tangent drawn at origin to the circle $S$ is

Let $P$ and $Q$ be two external points of the circle $S=x^2+y^2-a^2=0$. Let the chord of contact of the point $P$ with respect to the circle $S=0$ passes through $Q$. If $l_1$ and $l_2$ are the lengths of the tangents drawn from $P$ and $Q$ to the circle $S=0$, then $P Q=$

$A\left(x_1, y_1\right)$ is the internal centre of similitude and $B\left(x_2, y_2\right)$ is the external centre of similitude of two circles $C_1$ and $C_2$ whose centes are $P(\alpha, \beta)$ and $Q(\gamma, \delta)$, respectively. If $P A=3, A B=5, Q B=2$, then ratio of the radii of the two circles is

The equation of the direct common tangent of the circles $x^2+y^2-6 x-4 y-23=0$ and $x^2+y^2+2 x+2 y+1=0$ is

The length of the common chord of the two circles $x^2+y^2-4 x-8 y+4=0$ and $x^2+y^2-8 x-12 y+16=0$ is

If $A(1,1), B(-1,1)$ and $C(-1,-1)$ are three points and a point $P$ moves such that $(P A)^2=(P B)^2+(P C)^2$, then the equation of the locus of $P$ is

The radius of the circle passing through the points $(-1,1),(2,-1)$ and $(1,0)$ is

If $A=(0,-2)$ and $B$ is any point on the circle $x^2+y^2-2 x-2 y+1=0$, then the maximum value of $(\mathbf{A B})^2$ is

If $(\alpha, \beta)$ is the pole of the line $3 x-5 y+6=0$ with respect to the circle $x^2+y^2-10 x+14 y+46=0$, then $\alpha+\beta=$

$O(0,0)$ and $A(1,0)$ are centres of two units circles $C_1$ and $C_2$, respectively. $C_3$ is also a unit circle having its centre above $X$ - axis and passing through $O$ and $A$. The equation of the common tangent to $C_1$ and $C_3$ which does not intersect the circle $C_2$ is

If the circles $x^2+y^2-16 x-20 y+164=r^2(r>0)$ and $x^2+y^2-8 x-14 y+29=0$ intersect in two distinct points, then the maximum possible integral value of $r$ is

If the circle $x^2+y^2-6 x-12 y+1=0$ cuts another circle $C$ orthogonally and the centre of the circle $C$ is $(-4,2)$, then its radius of

The equation of the incircle of the triangle formed by the lines $x=0, y=0$ and $3 x+4 y-24=0$ is

If two tangents are drawn from the point $P\left(\frac{\pi}{4}\right)$ on the circle $x^2+y^2=4$ to the circle $x^2+y^2=1$, then the slopes of the tangents are

If $5 x+6 y-34=0$ and $2 x+y+c=0$ are conjugate lines with respect to the circle $x^2+y^2-8 x-10 y+25=0$, then the point on the line $2 x+y+c=0$ is

If $C_1$ and $C_2$ are the centres of similitude with respect to the circles $x^2+y^2+6 x+8 y+24=0$ and $x^2+y^2-6 x-8 y+9=0$, then $C_1 C_2=$

Let $x+y=0$ be the radical axis of the circles $S \equiv x^2+y^2+2 g x+2 f y+c=0$ and $S \equiv x^2+y^2-6 x-4 y+4=0$ and the radius of the circle $S=0$ be 1 . The $g+f=$

The radius of the circle which cuts all the three circles $x^2+y^2-4 x-4 y+3=0, x^2+y^2+4 x-4 y+3=0$ and $x^2+y^2+4 x+4 y+3=0$ orthogonally is

From a point $A(0,3)$ on the circle $(x+2)^2+(y-3)^2=4$, a chord $A B$ is drawn and it is extended to a point $Q$ such that $A Q=2 A B$. Then, the locus of $Q$ is

If $m_1, m_2$ are the slopes of the tangents drawn from a point $(1,-3)$ to the circle $x^2+y^2-6 x+4 y+12=0$, then $9\left(m_1^2+m_2^2\right)=$

If $A, B$ are the points of contact of the tangents drawn from the point $P(-2,-3)$ to the circle $x^2+y^2-8 x-10 y+5=0$ and the chord $A B$ subtends an angle $\theta$ at $P$, then $\tan \theta=$

The equation of the transverse common tangent of the circles $x^2+y^2-6 x-8 y+9=0$ and $x^2+y^2+2 x-2 y+1=0$

If $\theta$ is the angle between the circles

$x^2+y^2-2 x-4 y-4=0$ and $x^2+y^2-8 x-12 y+43=0$, then $|7 \sec \theta-18 \cos \theta|=$

If $\left(0, \frac{3}{4}\right)$ is the radical centre of the circles $S \equiv x^2+y^2+\alpha x+6 y=0, S \equiv x^2+y^2+2 \alpha x+\alpha y+6=0$ and $S^{\prime \prime} \equiv x^2+y^2+6 \alpha x-\alpha y+3=0$, then the distance between the radical centre and the centre of the circle $S^{\prime}=0$ is

Let the slope of a diameter $A C$ of a circle of radius 25 units be $\frac{3}{4}$. If $(3,2)$ is the centre of the circle, $A=\left(x_1, y_1\right)$ and $C=\left(x_2, y_2\right)$, then $\frac{x_1 x_2}{y_1 y_2}=$

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

A line drawn through the point $A(5,7)$ cut the circle $x^2+y^2-36=0$ at the points $P$ and $Q$. Then, $A P \cdot A Q=$

Let $P$ be any point on the circle $x^2+y^2-2 x-1=0$ and $C$ be its centre. Let $A B$ be the chord of contact of $P$ with respect to the circle $x^2+y^2-2 x=0$. Then, the locus of the circumcentre of the $\triangle C A B$ is

If a circle $C$ passing through $(4,0)$ touches the circle $x^2+y^2+4 x-6 y-12=0$ externally at the point $(1$, -1 ), then the radius of $C$ is

If the circles $C_1: x^2+y^2+2 x+4 y-20=0$, $C_2: x^2+y^2+6 x-8 y+9=0$ have $n$ common tangents and the length of the tangent drawn from the centre of similitude to the circle $C_2$ is $l$, then $\frac{l}{n^2}=$

If the common chord of the circles $x^2+y^2+4 y=0$ and $x^2+y^2-4 x-5=0$ is the diameter of the circle $S=0$, then the abscissa of the centre of the circle $S=0$ is

Let $a=1+i$ and $z=x+i y$. If the curve $z \bar{z}+a z+\bar{a} \bar{z}-4=0$ is cut by the straight line $(z+\bar{z})-i(z-\bar{z})+2=0$ at two points $A$ and $B$, then the equation of the circle passing through the origin, $A$ and $B$ is

A point $P$ moves so that distance from $(0,2)$ to $P$ is $\frac{1}{\sqrt{2}}$ times the distance of $P$ from $(-1,0)$. Then the locus of the point is

If $x^2+y^2-a^2+\lambda(x \cos \alpha+y \sin \alpha-p)=0$ is the smallest circle through the points of intersection of $x^2+y^2=a^2$ and $x \cos \alpha+y \sin \alpha=p, 0

If $P A$ and $P B$ are the tangents drawn from the point $P(1,1)$ to the circle $x^2+y^2+g x+g y-2=0$ with $C$ as the centre, then the area (in sq. units) of the quadrilateral $P A C B$ is

The point/points of intersection of the common tangents of the two circles $x^2+y^2-8 x-6 y+21=0$ and $x^2+y^2-2 y-15=0$ is/are

$L_1$ and $L_2$ are two common tangents to two circles. If $L_1$ touches the two circles at $A(1,1)$ and $B(0,1)$ and $L_2$ touches the two circles at $C\left(\frac{3}{5}, \frac{4}{5}\right), D\left(\frac{-1}{5}, \frac{7}{5}\right)$, then the equation of the radical axis of the two circles is

The centre of the smallest circle which cuts the circles $x^2+y^2-2 x-4 y-4=0$ and $x^2+y^2-10 x+12 y+52=0$ orthogonally is

If the parametric equations of the circle passing through the points $(3,4),(3,2)$ and $(1,4)$ is $x=a+r \cos \theta$, $y=b+r \sin \theta$, then $b^a r^a=$

From a point $P$ on the circle $x^2+y^2-4 x-6 y+9=0$, a pair of tangents $P Q$ and $P R$ are drawn touching the circle $x^2+y^2-4 x-6 y+12=0$ at $Q$ and $R$. If $C$ is the centre of the concentric circles, then the area of the $\triangle C Q R$ (in sq. units) is

The equations of the tangents drawn from the origin to the circle $x^2+y^2+2 g x+2 f y+g^2=0$ are

If $2 x+y=0$ is the equation of a chord of the circle $x^2+y^2-2 x-6 y+3=0$, then the circle with this chord as diameter passes through the point

If the radical axis of the circles $x^2+y^2+2 \alpha x+2 \beta y+c=0$ and $x^2+y^2+\frac{3}{2} x+4 y+c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$, then $4 \alpha \beta-8 \alpha-3 \beta+10=$

If the origin lies on a diameter of the circle $x^2+y^2-4 x-2 y-4=0$, then the equation of the circle passing through the end points of that diameter and the point $(1,2)$ is

If $\alpha \neq-4$ and $(2, \alpha)$ is the mid-point of a chord of the circle $x^2+y^2-4 x+8 y+6=0$, then the values of the $y$-intercept of the chord lie in the interval

$C_1$ and $C_2$ are the external and internal centres of similitude of the circles $x^2+y^2-2 x+4 y+1=0$ and $x^2+y^2+4 x-6 y+12=0$. If the radius of the circle having $C_1 C_2$ as its diameters is $r$, then $\frac{9}{2} r=$

Suppose the circle $S: x^2+y^2+2 g x+2 f y+c=0$ cuts orthogonally the two circles $S^{\prime}: x^2+y^2-4 x-6 y+11=0$ and $S^{\prime \prime}: x^2+y^2-10 x-4 y+21=0$. If the centre of $S=0$ lies on the bisector of the angle between the positive coordinate axes, then $2 g+2 f+c=$

If the circle $S_1: x^2+y^2=16$ intersects another circle $S_2$ of radius 5 units such that the common chord is of maximum length and slope $\frac{3}{4}$, then the centre of the circle $S_2$ is

If the polar of a point $P$ with respect to a circle of radius $r$ which touches the coordinate axes and lies in the first quadrant is $x+2 y=4 r$, then the point $P$ is

If the circles $x^2+y^2-2 x-2(3+\sqrt{7}) y+8+6 \sqrt{7}=0$ and $x^2+y^2-8 x-6 y+k^2=0, k \in \mathbf{Z}$, have exactly two common tangents, then the number of possible values of $k$ is

The circle $S=0$ cuts the circles

$C_1=x^2+y^2-8 x-2 y+16=0$ and $C_2=x^2+y^2-4 x-4 y-1=0$ orthogonally. If the common chord of $S=0$ and $C_1=0$ is $2 x+13 y-15=0$, then the centre of $S=0$ is

The equation of the circle passing through the points of intersection of the two orthogonal circles $S_1=x^2+y^2+k x-4 y-1=0$, $S_2=3 x^2+3 y^2-14 x+23 y-15=0$ and passing through the point $(-1,-1)$ is