2nd Year Maths IIB Study Material

What is a circle?

A circle is a locus of a point which moves in a plane, such that its distance from a fixed point in the plane is always constant.

The fixed point is called the centre of the circle and the constant is called its radius.

The equation of the circle: Its the set of all points lying on the circumference of the circle.

Chord and diameter of the circle: The line joining any two points on the circumference is called a chord. If any chord passing through its centre is called its diameter.

diameter d= 2r

Equation of the circle in different forms

(a). Centre- radius form : $\\ { \left( x-h \right) }^{ 2 }+{ \left( y-k \right) }^{ 2 }={ r }^{ 2 }$

This equation is known as central form of the equation of a circle.

NOTE: when centre (0, 0) then the equation of the circle becomes

$\\ { x }^{ 2 }+{ y}^{ 2 }={ r }^{ 2 }$

which is known as the standard form of the circle.

(b) Parametric form : The parametric equation of the circle $\\ { \left( x-h \right) }^{ 2 }+{ \left( y-k \right) }^{ 2 }={ r }^{ 2 }$ are

$\\ x=\quad h\quad +\quad r\quad cos\theta ,\quad y\quad =\quad k\quad +\quad r\quad sin\theta$

(c) General form: The equation of the circle with centre ( h, k) and radius r is

$\\ { \left( x-h \right) }^{ 2 }+{ \left( y-k \right) }^{ 2 }={ r }^{ 2 }$

Which is of the form ${ x }^{ 2 }+{ y }^{ 2 }$+2gx+2fy+c=0.