Circle
Circles
Circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane is constant.
Standard Forms of a Circle
(i) Equation of circle having centre $(h, k)$ and radius $r$ $$(xh)^{2}+(yk)^{2}=a^{2}$$
 If centre is $(0,0)$, then equation of circle is $$x^{2}+y^{2}=a^{2}$$
(ii) When the circle passes through the origin, then equation of the circle is $$x^{2}+y^{2}2 h x2 {ky}=0$$
(iii) When the circle touches the $\mathrm{X}$axis, the equation is $$\mathrm{x}^{2}+\mathrm{y}^{2}2 \mathrm{hx}2 \mathrm{ay}+\mathrm{h}^{2}=\mathrm{O}$$
(iv) Equation of the circle, touching the $Y$axis is $$x^{2}+y^{2}2 a x2 k y+k^{2}=0$$
(v) Equation of the circle, touching both axes is $$x^{2}+y^{2}2 a x2 a y+a^{2}=0$$
(vi) Equation of the circle passing through the origin and centre lying on the $\mathrm{X}$axis is
$$\mathrm{x}^{2}+\mathrm{y}^{2}2 \mathrm{ax}=0$$
(vii) Equation of the circle passing through the origin and centre lying on the Yaxis is $$x^{2}+y^{2} 2ay =0$$
(viii) Equation of the circle through the origin and cutting intercepts $a$ and $b$ on the coordinate axes is $$x^{2}+y^{2}ax by =0$$
(ix) Equation of the circle, when the coordinates of end points of a diameter are $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ and $\left(\mathrm{x} _{2}\right.,{y} _{2})$ is
$$\left(\mathrm{x}\mathrm{x} _{1}\right)\left(\mathrm{x}\mathrm{x} _{2}\right)+\left(\mathrm{y}\mathrm{y} _{1}\right)\left(\mathrm{y}\mathrm{y} _{2}\right)=0$$
(x) Equation of the circle passes through three given points $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right),\left(\mathrm{x} _{2}, \mathrm{y} _{2}\right)$ and $\left(\mathrm{x} _{3}, \mathrm{y} _{3}\right)$ is
$$ \left\begin{array}{llll} x^{2}+y^{2} & x & y & 1 \\ x_{1}^{2}+y_{1}^{2} & x_{1} & y_{1} & 1 \\ x_{2}^{2}+y_{2}^{2} & x_{2} & y_{2} & 1 \\ x_{3}^{2}+y_{3}^{2} & x_{3} & y_{3} & 1 \end{array}\right=0 . $$
(xi) Parametric equation of a circle
 For Circle $(\mathrm{x}\mathrm{h})^{2}+(\mathrm{y}\mathrm{k})^{2}=\mathrm{a}^{2}$
$$\mathrm{x}=\mathrm{h}+\mathrm{a} \cos \theta, \mathrm{y}=\mathrm{k}+\mathrm{a} \sin \theta,0 \leq \theta \leq 2 \pi$$
 For circle $\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{a}^{2}$ $$\mathrm{x}=\mathrm{a} \cos \theta, \mathrm{y}=\mathrm{a} \sin \theta,0 \leq \theta \leq 2 \pi$$
General Equation of a Circle
The general equation of a circle is given by $$x^{2}+y^{2}+2 g x+2 f y+c=0, $$ where centre of the circle $$C=(\mathrm{g},\mathrm{f})$$
and radius of the circle $$r=\sqrt{g^{2}+f^{2}c}$$

If $g^{2}+f^{2}c>0$, then the radius of the circle is real and hence the circle is also real.

If $g^{2}+f^{2}c=0$, then the radius of the circle is 0 and the circle is known as point circle.

If $g^{2}+f^{2}c<0$, then the radius of the circle is imaginary. Such a circle is imaginary, which is not possible to draw.
Position of a Point with Respect to a Circle
Let $(x_{1}, y_{1})$ be a point and a circle
$$ S \equiv x^{2} + y^{2} + 2gx + 2fy + c = 0 $$
 Inside $S_{1} < 0$
 Outside $S_{1} >0$
 On the circle $S_{1} =0$, where
$$ S_{1} = x_{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} + c $$
Intercepts on the Axes
(i) The length of the intercepts made by the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ with $X$ and $Y$axes are $2 \sqrt{g^{2}}\mathrm{c} \quad\text{and} \quad2 \sqrt{g^{2}}\mathrm{c}$
(ii) If $g^{2}>c$, then the roots of the equation $x^{2}+2 g x+c=0$ are real and distinct, so the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ meets the $X$axis in two real and distinct points.
(iii) If $g^{2}=c$, then the roots of the equation $x^{2}+2 g x+c=0$ are real and equal, so the circle touches $\mathrm{X}$axis, then intercept on $\mathrm{X}$axis is $\mathrm{O}$.
(iv) If $g^{2}<c$, then the roots of the equation $x^{2}+2 g x+c=0$ are imaginary, so the given circle does not meet $X$axis in real point. Similarly, the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ cuts the $\mathrm{Y}$axis in real and distinct points touches or does not meet in real point according to $\mathrm{f}^{2}>$, $=$ or $<\mathrm{c}$
Equation of Tangent
PYQ2023CircleQ3, PYQ2023CircleQ4
(i) Point Form

The equation of the tangent at the point $P\left(x_{1}, y_{1}\right)$ to a circle $x^{2}+y^{2} +2 g x+2 f y+c=0$ is $$\mathrm{xx} _{1}+\mathrm{yy} _{1}+\mathrm{g}\left(\mathrm{x}+\mathrm{x} _{1}\right)+\mathrm{f}\left(\mathrm{y}+\mathrm{y} _{1}\right)+\mathrm{c}=0$$

The equation of the tangent at the point $P\left(x_{1}, y_{1}\right)$ to a circle $x^{2}+y^{2}$ is $$x x_{1}+y y_{1}=r^{2}$$
(ii)Slope Form

The equation of the tangent of slope $m$ to the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ are $$y+f=m(x+g) \pm \sqrt{ }\left(g^{2}+f^{2}c\right)\left(1+m^{2}\right)$$

The equation of the tangents of slope $m$ to the circle $(xa)^{2}(yb)^{2}=r^{2}$ are $yb=m(x$ a) $\pm \mathrm{r} \sqrt{ }\left(1+\mathrm{m}^{2}\right)$ and the coordinates of the points of contact are
$$ \left(a \pm \frac{m r}{\sqrt{1+m^{2}}}, b \mp \frac{r}{\sqrt{1+m^{2}}}\right) $$
 The equation of tangents of slope $m$ to the circle $x^{2}+y^{2}=r^{2}$ are $y=m x \pm r \sqrt{ }\left(1+m^{2}\right)$ and the coordinates of the point of contact are
$$ \left( \pm \frac{r m}{\sqrt{1+m^{2}}}, \mp \frac{r}{\sqrt{1+m^{2}}}\right) $$
(iii)Parametric Form
The equation of the tangent to the circle $(xa)^{2}+(yb)^{2}=r^{2}$ at the point $(a+r \cos \theta, b+r$ $\sin \theta)$ is $$(xa) \cos \theta+(yb) \sin \theta=r$$
Equation of Normal
(i) Point Form
 The equation of normal at the point $\left(x_{1}, y_{1}\right)$ to the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ is $$y  y_{1} = (\frac{y_{1} + f}{x_{1} + g})(x  x_{1})$$
i.e. $$\left(\mathrm{y} _{1}+\mathrm{f}\right) \mathrm{x}\left(\mathrm{x} _{1}+\mathrm{g}\right) \mathrm{y}+\left(\mathrm{gy} _{1}\mathrm{fx} _{1}\right)=0$$
 The equation of normal at the point $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{r}^{2}$ is $$\frac{x}{x _{1}} = \frac{y}{y _{1}}$$
(ii) Parametric Form
The equation of normal to the circle $x^{2}+y^{2}=r^{2}$ at the point $(r \cos \theta, r \sin \theta)$ is $$\frac{x}{r} \cos \theta= \frac{y}{r} \sin \theta \quad \text{or} \quad \mathrm{y}=\mathrm{x} \tan \theta$$
Important Points to be Remembered
(i) The line $y=m x+c$ meets the circle in unique real point or touch the circle $\mathrm{x}^2+\mathrm{y}^{2}=\mathrm{r}^2$, if $$r = \left \frac{c}{\sqrt{1 + m^2}} \right$$
and the point of contacts are
$$ \left(\frac{ \pm m r}{\sqrt{1+m^{2}}}, \frac{\mp r}{\sqrt{1+m^{2}}}\right) $$
(ii) The line $\mathrm{lx}+\mathrm{my}+\mathrm{n}=0$ touches the circle $x^{2}+y^{2}=r^{2}$, if $$\mathrm{r}^{2}\left(\mathrm{l}^{2}+\mathrm{m}^{2}\right)=\mathrm{n}^{2}$$
(iii) Tangent at the point $P(\theta)$ to the circle $x^{2}+y^{2}=r^{2}$ is $$x \cos \theta+y \sin \theta=r$$
(iv) The point of intersection of the tangent at the points $\mathrm{P}\left(\theta_{1}\right)$ and $\mathrm{Q}\left(\theta_{2}\right)$ on the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=$ $r^{2}$
$$x = \frac{r \cos \left(\frac{\theta_{1} + \theta_{2}}{2}\right)}{\cos \left(\frac{\theta_{1}  \theta_{2}}{2}\right)} \quad \text{and} \quad y = \frac{r \sin \left(\frac{\theta_{1} + \theta_{2}}{2}\right)}{\cos \left(\frac{\theta_{1}  \theta_{2}}{2}\right)}$$
(v) Normal at any point on the circle is a straight line which is perpendicular to the tangent to the curve at the point and it passes through the centre of circle.
(vi) Power of a point $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ with respect to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{c}=0$ is $$\mathrm{x} _{1}{ }^{2}+\mathrm{y} _{1}{ }^{2}+2 \mathrm{gx} _{1}+2 \mathrm{fy} _{1}+\mathrm{c}$$
(vii) If $\mathrm{P}$ is a point and $\mathrm{C}$ is the centre of a circle of radius $\mathrm{r}$, then the maximum and minimum distances of $\mathrm{P}$ from the circle are $\mathrm{CP}+\mathrm{r}$ and $\mathrm{CP}\mathrm{r}$ respectively.
(viii) If a line is perpendicular to the radius of a circle at its end points on the circle, then the line is a tangent to the circle and viceversa.
Pair of Tangents
PYQ2023CircleQ5 , PYQ2023CircleQ9
(i) The combined equation of the pair of tangents drawn from a point $\mathrm{P}\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ to the circle $\mathrm{x}^{2}+$ $y^{2}=r^{2}$ is
$$\left(x^{2}+y^{2}r^{2}\right)\left(x _{1}^{2}+y _{1}^{2}r _{1}^{2}\right)=\left(x _{1}+y _{1}r^{2}\right)^{2}$$
or $$\mathrm{SS} _{1}=\mathrm{T}^{2}$$
where $$S=x^{2}+y^{2}r^{2}, S _{1}=x _{1}{ }^{2}+y _{1}{ }^{2}r _{1}{ }^{2}$$
and $$\mathrm{T}=\mathrm{xx} _{1}+\mathrm{yy} _{1}\mathrm{r}^{2}$$
(ii) The length of the tangents from the point $P\left(x _{1}, y _{1}\right)$ to the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ is equal to
$$ \sqrt{x _{1}^{2}+y _{1}^{2}+2 g x _{1}+2 f y _{1}+c}=\sqrt{S _{1}} $$
(iii) Chord of contact TT’ of two tangents drawn from $\mathrm{P}\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$

For the Circle $\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{r}^{2}$ $$\mathrm{T}=0$$

For the Circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ is
$$x_{1}+y_{1}+g\left(x+x_{1}\right)+f\left(y+y_{1}\right)+c=0$$
(iv) Equation of Chord Bisected at a Given Point The equation of chord of the circle $\mathrm{S} \equiv$ $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{c}=0$ bisected at the point $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ $$\mathrm{T}=\mathrm{S} _{1},\quad\text{where}\quad S _1=\mathrm{x} _{1}^{2}+\mathrm{y} _{1}^{2}+2 \mathrm{gx} _{1}+\mathrm{fy} _{1}+\mathrm{c}$$
i.e., $$T=\mathrm{xx} _{1}+\mathrm{yy} _{1}+\mathrm{g}\left(\mathrm{x}+\mathrm{x} _{1}\right)+\mathrm{f}\left(\mathrm{y}+\mathrm{y} _{1}\right)+\mathrm{c}$$
(v) Director Circle The locus of the point of intersection of two perpendicular tangents to a given circle is called a director circle. For circle $x^{2}+y^{2}=r^{2}$,
the equation of director circle is $$x^{2}+y^{2}=2 r^{2}$$
Common Chord
The chord joining the points of intersection of two given circles is called common chord.
(i) If $S _{1}=0$ and $S _{1}=0$ be two circles, such that
$$S_{1} \equiv x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0, S_{2} \equiv x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$$
then their common chord is given by $$\mathrm{S} _{1}\mathrm{S} _{2}=0$$
(ii) If $\mathrm{C} _{1}, \mathrm{C} _{2}$ denote the centre of the given circles, then their common chord
$$PQ = 2PM = 2\sqrt{(C _{1}P)^{2}  (C _{1}M)^{2}}$$
(iii) If $r_{1}$, and $r_{2}$ be the radii of ’two circles, then length of common chord is $$\frac{2 r_{1} r_{2}}{\sqrt{r_{1}^{2}+r_{2}^{2}}}$$
Angle of Intersection of Two Circles
The angle of intersection of two circles is defined as the angle between the tangents to the two circles at their point of intersection is given by
$$\cos \theta = \frac{r_{1}^{2} + r_{2}^{2}  d^{2}}{2 r_{1} r_{2}}$$
Orthogonal Circles
Two circles are said to be intersect orthogonally, if their angle of intersection is a right angle.
If two circles $\mathrm{S} _{1} \equiv \mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{~g} _{1} \mathrm{x}+2 \mathrm{f} _{1} \mathrm{y}+\mathrm{c} _{1}=0$ and $\mathrm{S} _{2} \equiv \mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{~g} _{2} \mathrm{x}+2 \mathrm{f} _{2} \mathrm{y}+\mathrm{c} _{2}=0$ are orthogonal, then $$2 \mathrm{~g} _{1} \mathrm{~g} _{2}+2 \mathrm{f} _{1} \mathrm{f} _{2}=\mathrm{c} _{1}+\mathrm{c} _{2}$$
Family of Circles
(i) The equation of a family of circles passing through the intersection of a circle $x^{2}+y^{2}+2 g x$ $+2 \mathrm{fy}+\mathrm{c}=0$ and line $\mathrm{L}=1 \mathrm{x}+\mathrm{my}+\mathrm{n}=0$ is $$\mathrm{S}+\lambda \mathrm{L}=0$$
where $ \lambda $ is any real number.
(ii) The equation of the family of circles passing through the point $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{1}, y_{1}\right)$ is $$\left(xx_{1}\right)\left(xx_{2}\right)+\left(yy_{1}\right)\left(yy_{2}\right)+\lambda\left\begin{array}{lll}x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1\end{array}\right=0$$
(iii) The equation of the family of circles touching the circle $S \equiv x^{2} + y^{2} + 2gx + 2fy + c = 0$ at point $P(x_{1}, y_{1})$ is
$$x^{2} + y^{2} + 2gx + 2fy + c + \lambda\left[x x_{1} + y y_{1} + g(x + x_{1}) + f(y + y_{1}) + c\right] = 0$$
or $$S + \lambda L = 0$$ where $L = 0$ is the equation of the tangent to $S = 0$ at $(x_{1}, y_{1})$ and $\lambda \in \mathbb{R}$
(iv) Any circle passing through the point of intersection of two circles $S_{1}$ and $S_{2}$ is
$$S_{1} + \lambda\left(S_{1}  S_{2}\right) = 0$$
Radical Axis
(i) The radical axis of two circles is the locus of a point which moves in such a way that the length of the tangents drawn from it to the two circles are equal.
(ii) The radical axis of two circles $S_{1}=0$ and $S_{2}=0$ is given by $$S_{1}S_{2}=0$$
(iii) The radical axis of two circles is always perpendicular to the line joining the centres of the circles.
(iv) The radical axis of three vertices, whose centres are noncollinear taken in pairs of concurrent.
(v) The centre of the circle cutting two given circles orthogonally, lies on their radical axis.
(vi) Radical Centre The point of intersection of radical axis of three circles whose centre are noncollinear, taken in pairs, is called their radical centre.
Pole and Polar
(i) If through a point $\mathrm{P}\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ (within or outside a circle) there be drawn any straight line to meet the given circle at $\mathrm{Q}$ and $\mathrm{R}$, the locus of the point of intersection of tangents at $\mathrm{Q}$ and $\mathrm{R}$ is called the polar of $\mathrm{P}$ and polar at $\mathrm{P}$ is called the pole of polar.
(ii) Equation of polar to the circle $x^{2}+y^{2}=r^{2}$ is $$x x_{1}+y y_{1}=r^{2}$$
(iii) Equation of polar to the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ is $$x x_{1} + y y_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0$$
(iv) Conjugate Points: Two points $\mathrm{A}$ and $\mathrm{B}$ are conjugate points with respect to a given circle, if each lies on the polar of the other with respect to the circle.
(v) Conjugate Lines: Conjugate If two lines be such that the pole of one lies on the other, then they are called conjugate lines with respect to the given circle.
Coaxial System of Circles
A system of circle is said to be coaxial system of circles, if every pair of the circles in the system has same radical axis.The equation of a system of coaxial circles, when the equation of the radical axis $P \equiv lx$ $+m y+n=0$ and one of the circle of the system $S=x^{2}+y^{2}+2 g x+2 f y+c=0$ is $$S+\lambda \mathrm{P}=0$$
Since, the lines joining the centres of two circles is perpendicular to their radical axis. Therefore, the centres of all circles of a coaxial system lie on a straight line, which is perpendicular to the common radical axis.
Important Points to be Remembered
(i) Circle touching a line $\mathrm{L}=\mathrm{O}$ at a point $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ on it is
$$\left(\mathrm{x}\mathrm{x} _{1}\right)^{2}+\left(\mathrm{y}\mathrm{y} _{1}\right)^{2}+\mathrm{L}=0$$
(ii) Circumcircle of a A with vertices $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right),\left(\mathrm{x} _{2}, \mathrm{y} _{2}\right),\left(\mathrm{x} _{3}, \mathrm{y} _{3}\right)$ is
$$\frac{(x  x_{1})(x  x_{2}) + (y  y_{1})(y  y_{2})}{(x_{3}  x_{1})(x_{3}  x_{2}) + (y_{3}  y_{1})(y_{3}  y_{2})}$$
or
$$\frac{\left\begin{array}{ccc} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right}{\left\begin{array}{ccc} x_{3} & y_{3} & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right}$$
(iii) A line intersect a given circle at two distinct real points, if the length of the perpendicular from the centre is less than the radius of the circle.
(iv) Length of the intercept cut off from the line $y=m x+c$ by the circle $x^{2}+y^{2}=a^{2}$ is $$2 \sqrt{\frac{a^{2}\left(1+m^{2}\right)c^{2}}{1+m^{2}}}$$
(v) In general, two tangents can be drawn to a circle from a given point in its plane. If $\mathrm{m} _{1}$ and $m _{2}$ are slope of the tangents drawn from the point $P\left(x _{1}, y _{1}\right)$ to the circle $x^{2}+y^{2}=a^{2}$, then
$$ m_{1}+m_{2}=\frac{2 x_{1} y_{1}}{x_{1}^{2}a^{2}} \text { and } m_{1} \times m_{2}=\frac{y_{1}^{2}a^{2}}{x_{1}^{2}a^{2}} $$
(vi) Pole of $\mathrm{lx}+\mathrm{my}+\mathrm{n}=0$ with respect to $\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{a}^{2}$ is
$$ \left(\frac{a^{2} l}{n},\frac{a^{2} m}{n}\right) $$
(vii) Let $S_{1}=0, S_{2}=0$ be two circles with radii $r_{1}, r_{2}$, then $$\frac{S_{1}}{r_{1}} \pm \frac{S_{2}}{r_{2}} = 0$$ will meet at right angle.
(viii) The angle between the two tangents from $\left(x_{1}, y_{1}\right)$ to the circle $x^{2}+y^{2}=a^{2}$ is
$$2 \tan^{1}\left(\frac{a}{\sqrt{S_{1}}}\right)$$
(ix) The pair of tangents from $(0,0)$ to the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ are at right angle if $\mathrm{g}^{2}+\mathrm{f}^{2}=2 \mathrm{c}$
(x) If $\left(x_{1}, y_{1}\right)$ is one end of a diameter of the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$, then the other end will be $\left(2 \mathrm{~g}\mathrm{x} _{1},2 \mathrm{f}\mathrm{y} _{1}\right)$
Image of the Circle by the Line Minor
Let the circle be $x^{2}+y^{2}+2 g x+2 f y+c=0$ and line minor $\mathrm{l} \mathrm{x}+\mathrm{my}+\mathrm{n}=0$ then the image of the circle is
$$\left(\mathrm{x}\mathrm{x} _{1}\right)^{2}+\left(\mathrm{y}\mathrm{y} _{1}\right)^{2}=\mathrm{r}^{2}$$
where, $r=\sqrt{ } g^{2}+f^{2}c$
Diameter of a Circle
The locus of the middle points of a system of parallel chords of a circle is called a diameter of the circle.
(i) The equation of the diameter bisecting parallel chords $y=m x+c$ of the circle $x^{2}+y^{2}=a^{2}$ is $$\mathrm{x}+\mathrm{my}=0$$
(ii) The diameter corresponding to a system of parallel chords of a circle always passes through the centre of the circle and is perpendicular to the parallel chords.
Common Tangents of Two Circles
Let the centres and radii of two circles be $C_{1}, C_{2}$ and $r_{1}, r_{2}$, respectively.
(i) When one circle contains another circle, no common tangent is possible.
Condition: $$C_{1}C_{2} < r_{1}  r_{2}$$
(ii) When two circles touch internally, one common tangent is possible.
Condition: $$C_{1}C_{2} = r_{1}  r_{2}$$
(iii) When two circles intersect, two common tangents are possible.
Condition: $$r_{1}  r_{2} < C_{1}C_{2} < r_{1} + r_{2}$$
(iv) When two circles touch externally, three common tangents are possible.
Condition: $$C_{1}C_{2} = r_{1} + r_{2}$$
(v) When two circles are separate, four common tangents are possible.
Condition: $$C_{1}C_{2} > r_{1} + r_{2}$$
Important Points to be Remembered
Let AS be a chord of contact of tangents from $C$ to the circle $x^{2} + y^{2} = r^{2}$. M is the midpoint of AB.