**Section I:**

**Definition of a Circle:**

A circle is the locus of a point which moves in a plane so that it’s distance from a *fixed point* in the plane is always *constant*.The fixed point is called the *centre* of the circle and the constant distance is called its *radius*.

**Section II:**

**Equations of a circle:**

- An equation of a circle with centre and radius r is .
- An equation of a circle with centre and radius r is .
- An equation of a circle on the line segment joining and as diameter is .
- General equation of a circle is : where g, f, and c are constants
- centre of this circle is
- Its radius is ,
- Length of the intercept made by this circle on the x-axis is if and that on the y-axis is if .

- General equation of second order degree in x and y represents a circle if and only if:
- coefficient of equals coefficient of , that is,
- coefficient of is zero, that is ,

**Section III: Some results regarding circles:**

*Position of a point with respect to a circle:*Point lies outside, on or inside a circle , according as*Parametric coordinates*of any point on the circle are given by with . In particular, parametric coordinates of any point on the circle.- An equation of the
*tangent*to the circle at the point on the circle is - An equation of the
*normal*to the circle at the point on the circle is - Equations of the
*tangent*and*normal*to the circle at the point on the circle are, respectively, and - The line is a tangent to the circle if and only if .
- The lines are tangents to the circle , for all finite values of m. If m is infinite, the tangents are .
- An equation of the
*chord*of the circle , whose mid-point is is , where and . In particular, an equation of the chord of the circle , whose mid-point is is . - An equation of the
*chord of contact*of the tangents drawn from a point outside the circle is .(S and T are as defined in (8) above). *Length of the tangent*drawn from a point outside the circle , to the circle, is . (S and ) are as defined in (8) above.)- Two circles with centres and and radii , respectively, (i)
*touch each other externally*if . the point of contact is and (ii)*touch each other internally*if , where ; the point of contact is - An equation of the
*family of circles*passing through the points and is , where

- An equation of the family of circles which touch the line at for any finite value of m is . If m is infinite, the equation becomes .
- Let QR be a chord of a circle passing through the point and let the tangents at the extremities Q and R of this chord intersect at the point . Then, locus of L is called the
of P with respect to the circle, and P is called the**polar****pole of its polar.***Equation of the polar of*with respect to the circle is , where T is defined as above.- If the polar of P with respect to a circle passes through Q, then the polar of Q with respect to the same circle passes through P. Two such points P and Q, are called
**conjugate points***of the same circle.*

- If lengths of the tangents drawn from a point P to the two circles and are equal, then the locus of P is called the
of the two circles and , and its equation is , that is,*radical axis*- Radical axis of two circles is perpendicular to the line joining their circles.
- Radical axes of three circles, taken in pairs, pass through a fixed point called the
of the three circles, if the centres of these circles are non-collinear.*radical centre*

**4: Special Forms of Equation of a Circle:**

- An equation of a circle with centre and radius is . This touches the co-ordinate axes at the points and .
- An equation of a circle with centre , radius is . This touches the x-axis at .
- An equation of a circle with centre and radius is . This circle passes through the origin , and has intercepts a and b on the x and y axes, respectively.

**5: Systems of Circles:**

Let ; and and .

- If two circles and intersect at real and distinct points, then where represents a family of circles passing through these points, where is a parameter, and when represents the chord of the circles.
- If two circles and touch each other, then represents equation of the common tangent to the two circles at their point of contact.
- If two circles and intersect each other
*orthogonally*(the tangents at the point of intersection of the two circles are at right angles), then . - If the circle intersects the line at two real and distinct points, then represents a family of circles passing through these points.
- If is a tangent to the circle at P, then represents a family of circles touching at P, and having as the common tangent at P.
*Coaxial Circles:*A system of circles is said to be coaxial if every pair of circles of the system have the same radical axis. The*simplest form*of the equation of a coaxial system of circles is : , where g is a variable and c is constant, the common radical axis of the system being y-axis and the line of centres being x-axis.*The Limiting points of the coaxial system of circles*are the members of the system which are of zero radius. Thus, the limiting points of the coaxial system of circles are if . The equation () represents a family of coaxial circles, two of whose members are given to be and .*Conjugate systems (or orthogonal systems) of circles :*Two system of circles such that every circle of one system cuts every circle of the other system orthogonally are said to be conjugate system of circles. For instance, and , where g and f are variables and c is constant, represent two systems of coaxial circles which are conjugate.

**6: Common tangents to two circles:**

If and are two circles with centres and and radii and respectively, then we have the following results regarding their common tangents:

- When , that is, distance between the centres is greater than the sum of their radii, the two circles do not intersect with each other, and four common tangents can be drawn to circles. Two of them are direct common tangents and other two are transverse common tangents. The points of intersection of direct common tangents and transverse common tangents respectively, always lie on the line joining the centres of the two circles and divide it externally and internally respectively in the ratio of their radii.
- When , that is, the distance between the centres is equal to the sum of their radii, the two circles touch each other externally, two direct tangents are real and distinct and the transverse tangents coincide.
- When , that is, the distance between the centres is less than the sum of the radii, the circles intersect at two real and distinct points, the two direct common tangents are real and distinct while the transverse common tangents are imaginary.
- When with , that is, the distance between the centres is equal to the difference of their radii, the circles touch each other internally, two direct common tangents are real and coincident, while the transeverse common tangents are imaginary.
- When , with , that is, the distance between the centres is less than the difference of the radii, one circle with smaller radius lies inside the other and the four common tangents are all imaginary.

*To be continued,*

*Nalin Pithwa.*