For $$\mathrm{n}\in \mathbf{N}$$, let $${\mathrm{S}}_{\mathrm{n}}=\left\{z\in \mathbf{C}:|z-3+2i|=\frac{\mathrm{n}}{4}\right\}$$ and $${\mathrm{T}}_{\mathrm{n}}=\left\{z\in \mathbf{C}:|z-2+3i|=\frac{1}{\mathrm{n}}\right\}$$. Then the number of elements in the set $$\left\{n\in \mathbf{N}:S_n\cap T_n=\varphi \right\}$$ is :

For $$z\in \mathbb{C}$$ if the minimum value of $$(|z-3\sqrt{2}|+|z-p\sqrt{2}i|)$$ is $$5\sqrt{2}$$, then a value Question: of $$p$$ is _____________.

Let O be the origin and A be the point $$z_1=1+2i$$. If B is the point $$z_2$$, $${\mathop{\rm Re}\nolimits} ({z_2})

If $$z=x+iy$$ satisfies $$|z|-2=0$$ and $$|z-i|-|z+5i|=0$$, then :

Let the minimum value $$v_0$$ of $$v=|z{|}^2+|z-3{|}^2+|z-6i{|}^2,z\in \mathbb{C}$$ is attained at $$z=z_0$$. Then $${\left|2z_0^2-{\bar{z}}_0^3+3\right|}^2+v_0^2$$ is equal to :

Let S be the set of all $$($\alpha$, $\beta$), $\pi$

Let $$S_1=\left\{z_1\in \mathbf{C}:\left|z_1-3\right|=\frac{1}{2}\right\}$$ and $$S_2=\left\{z_2\in \mathbf{C}:\left|z_2-\right|z_2+1||=\left|z_2+\right|z_2-1||\right\}$$. Then, for $$z_1\in S_1$$ and $$z_2\in S_2$$, the least value of $$\left|z_2-z_1\right|$$ is :

If $$z=2+3i$$, then $$z^5+(\bar{z}{)}^5$$ is equal to :

If $$z\ne 0$$ be a complex number such that $$\left|z-\frac{1}{z}\right|=2$$, then the maximum value of $$|z|$$ is :

Let $$\mathrm{S}=\{z=x+i y:|z-1+i| $\ge$|z|,|z|

Let the tangent to the circle C1 : x2 + y2 = 2 at the point M($$-$$1, 1) intersect the circle C2 : (x $$-$$ 3)2 + (y $$-$$ 2)2 = 5, at two distinct points A and B. If the tangents to C2 at the points A and B intersect at N, then the area of the triangle ANB is equal to :

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

If the tangents drawn at the points $$O(0,0)$$ and $$P\left(1+\sqrt{5},2\right)$$ on the circle $$x^2+y^2-2x-4y=0$$ intersect at the point Q, then the area of the triangle OPQ 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, - {1 \over 3}} \right) lies on or inside the circle C, is :

Let C be a circle passing through the points A(2, $$-$$1) and B(3, 4). The line segment AB s 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} = {{13} \over 2}, then r2 is equal to :

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

Let a circle C touch the lines $$L_1:4x-3y+K_1=0$$ and $$L_2=4x-3y+K_2=0$$, $$K_1,K_2\in R$$. If a line passing through the centre of the circle C intersects L1 at $$(-1,2)$$ and L2 at $$(3,-6)$$, then the equation of the circle C is :

Consider three circles: $$C_1:x^2+y^2=r^2$$ $$C_2:(x-1{)}^2+(y-1{)}^2=r^2$$ $$C_3:(x-2{)}^2+(y-1{)}^2=r^2$$ If a line L : y = mx + c be a common tangent to C1, C2 and C3 such that C1 and C3 lie on one side of line L while C2 lies on other side, then the value of $$20(r^2+c)$$ is equal to :

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 $$\mathrm{P}$$ and $$\mathrm{Q}$$ be the roots of $$y^2+2y-7=0$$. If $$\mathrm{PQ}$$ is a diameter of the circle $$x^2+y^2+2ax+2by+c=0$$, then the value of $$(a+b-c)$$ is _____________.

If the circle $$x^2+y^2-2gx+6y-19c=0,g,c\in \mathbb{R}$$ passes through the point $$(6,1)$$ and its centre lies on the line $$x-2cy=8$$, then the length of intercept made by the circle on $$x$$-axis is :