Let $$z$$ be a complex number such that $$|z+2|=1$$ and $$\mathrm{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$$. Then the value of $$|\mathrm{Re}(\overline{z+2})|$$ is

If the set $$R=\{(a,b):a+5b=42,a,b\in \mathbb{N}\}$$ has $$m$$ elements and \sum_\limits{n=1}^m\left(1-i^{n !}\right)=x+i y, where $$i=\sqrt{-1}$$, then the value of $$m+x+y$$ is

Consider the following two statements : Statement I: For any two non-zero complex numbers $$z_1,z_2,(|z_1|+|z_2|)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right|\le 2\left(\left|z_1\right|+\left|z_2\right|\right)\text{, and }$$ Statement II : If $$x,y,z$$ are three distinct complex numbers and $$\mathrm{a},\mathrm{b},\mathrm{c}$$ are three positive real numbers such that $$\frac{\mathrm{a}}{|y-z|}=\frac{\mathrm{b}}{|z-x|}=\frac{\mathrm{c}}{|x-y|}$$, then $$\frac{{\mathrm{a}}^2}{y-z}+\frac{{\mathrm{b}}^2}{z-x}+\frac{{\mathrm{c}}^2}{x-y}=1$$. Between the above two statements,

Let $$S_1=\{z\in \mathbf{C}:|z|\le 5\},S_2=\left\{z\in \mathbf{C}:\mathrm{Im}\left(\frac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right)\ge 0\right\}$$ and $$S_3=\{z\in \mathbf{C}:\mathrm{Re}(z)\ge 0\}$$. Then the area of the region $$S_1\cap S_2\cap S_3$$ is :

If $$z_1,z_2$$ are two distinct complex number such that $$\left|\frac{z_1-2z_2}{\frac{1}{2}-z_1{\bar{z}}_2}\right|=2$$, then

Let the locus of the midpoints of the chords of the circle $x^2+(y-1{)}^2=1$ drawn from the origin intersect the line $x+y=1$ at $\mathrm{P}$ and $\mathrm{Q}$. Then, the length of $\mathrm{PQ}$ is :

Let $C:x^2+y^2=4$ and $C^{\prime }:x^2+y^2-4\lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $\mathrm{C}$ and $\mathrm{C}$ intersect at two distinct points, is $\mathrm{R}-[\mathrm{a},\mathrm{b}]$, then the point $(8\mathrm{a}+12,16\mathrm{b}-20)$ lies on the curve :

Four distinct points $(2 k, 3 k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to :

Let a variable line passing through the centre of the circle $$x^2+y^2-16x-4y=0$$, meet the positive co-ordinate axes at the points $$A$$ and $$B$$. Then the minimum value of $$OA+OB$$, where $$O$$ is the origin, is equal to

If one of the diameters of the circle $$x^2+y^2-10x+4y+13=0$$ is a chord of another circle $$\mathrm{C}$$, whose center is the point of intersection of the lines $$2x+3y=12$$ and $$3x-2y=5$$, then the radius of the circle $$\mathrm{C}$$ is :

If the circles $$(x+1{)}^2+(y+2{)}^2=r^2$$ and $$x^2+y^2-4x-4y+4=0$$ intersect at exactly two distinct points, then

Let a circle passing through $$(2,0)$$ have its centre at the point $$(\mathrm{h},\mathrm{k})$$. Let $$(x_{\mathrm{c}},y_{\mathrm{c}})$$ be the point of intersection of the lines $$3x+5y=1$$ and $$(2+\mathrm{c})x+5{\mathrm{c}}^{2}y=1$$. If \mathrm{h}=\lim _\limits{\mathrm{c} \rightarrow 1} x_{\mathrm{c}} and \mathrm{k}=\lim _\limits{\mathrm{c} \rightarrow 1} y_{\mathrm{c}}, then the equation of the circle is :

A square is inscribed in the circle $$x^2+y^2-10x-6y+30=0$$. One side of this square is parallel to $$y=x+3$$. If $$\left(x_i,y_i\right)$$ are the vertices of the square, then $$\Sigma \left(x_i^2+y_i^2\right)$$ is equal to:

Let $$\mathrm{C}$$ be a circle with radius $$\sqrt{10}$$ units and centre at the origin. Let the line $$x+y=2$$ intersects the circle $$\mathrm{C}$$ at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$. Let $$\mathrm{MN}$$ be a chord of $$\mathrm{C}$$ of length 2 unit and slope $$-1$$. Then, a distance (in units) between the chord PQ and the chord $$\mathrm{MN}$$ is

If the image of the point $$(-4,5)$$ in the line $$x+2y=2$$ lies on the circle $$(x+4{)}^2+(y-3{)}^2=r^2$$, then $$r$$ is equal to:

Let the circles $$C_1:(x-\alpha {)}^2+(y-\beta {)}^2=r_1^2$$ and $$C_2:(x-8{)}^2+{\left(y-\frac{15}{2}\right)}^2=r_2^2$$ touch each other externally at the point $$(6,6)$$. If the point $$(6,6)$$ divides the line segment joining the centres of the circles $$C_1$$ and $$C_2$$ internally in the ratio $$2:1$$, then $$(\alpha +\beta )+4\left(r_1^2+r_2^2\right)$$ equals

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point $$(3,2)$$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $$(5,5)$$ is :

Let the circle $$C_1:x^2+y^2-2(x+y)+1=0$$ and $${\mathrm{C}}_2$$ be a circle having centre at $$(-1,0)$$ and radius 2 . If the line of the common chord of $${\mathrm{C}}_1$$ and $${\mathrm{C}}_2$$ intersects the $$\mathrm{y}$$-axis at the point $$\mathrm{P}$$, then the square of the distance of P from the centre of $${\mathrm{C}}_1$$ is:

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies :

If $$\mathrm{P}(6,1)$$ be the orthocentre of the triangle whose vertices are $$\mathrm{A}(5,-2),\mathrm{B}(8,3)$$ and $$\mathrm{C}(\mathrm{h},\mathrm{k})$$, then the point $$\mathrm{C}$$ lies on the circle :