Mathematics Revision Notes of Circle for NDA

Circle :

The locus of a point which moves in such a way that its distance from a fixed point in the plane remains constant, is called a circle. This fixed point, is called the centre and the constant distance, is called the radius of the circle.

General Form of Equation of Circle

(i) The general form of the equation of a circle is `x ^2 + y^2 + 2gx + 2fy + c = 0`

Its centre is `(- g,- f)` and radius is `sqrt (g^2 + f^2 -c)`

(ii) The general form of the equation of a circle can also be written as

`ax^2 + ay^2 + 2gx + 2fy +c=0, a ne 0`

Its centre is `((-g)/a , (-f)/a)` and radius is `sqrt(g^2/a^2 +f^2/a^2 - c/a)`

Different Types of Equations of Circle

(i) Circle having Centre `(h, k)` and Radius `r`

Equation of the circle is `(x -h)^2 + (y- k )^2 = r^2`



(ii) Circle having Centre `(0,0)` and Radius `r`

Equation of the circle is `x ^2 + y^2 = r ^2`


(iii) `text(Centre at X -axis)` y-coordinate of

centre `= 0`, i.e. `f = 0`

Centre `=(-g,0)`

Equation of the circle is

`x^2 + y^2 + 2gx + c = 0`


(iv) `text(Centre at Y-axis)` x-coordinate of centre `= 0`, i.e. `g = 0`

Centre `= (0,- f)`

Equation of the circle is

`x^2 + y^2 + 2fy + c = 0`



(v) `text(Centre at Origin)`

Centre `= (-g, -f)= (0, 0)`

Equation of the circle is `x^2 + y^2+ c = 0`

(vi) `text(Circle Passing through Origin)`

Equation of the circle is

`x ^2 + y^2 + 2gx + 2fy = 0`

because `(0, 0)` should satisfy the equation of the circle,

i.e. `c = 0`



(vii) `text(Circle Touching Y-axis at Origin and Centred at X-axis)`

Equation of the circle is

`x ^2 + y^2 -2gx =0`


(viii) `text(Circle Touching X -axis a Origin and Centred at Y-axis)`

Equation of the circle is `x^ 2 + y^2- 2fy = 0`

`(0, 0)` and `(0, -2 f)` are the end points of the diameter.


(ix) `text(Equation of a Circle in Diametric Form)` If `(x_1,y_1)` and `(x_2, y_2 )` are the end points of one of the diameter, then the equation of the circle is

`(x- x_1) (x- x_2) + (y- y_1) (y- y_2) = 0`


(x) `text(Parametric Form of a Circle)`

Consider the circle `(x - h )^2 + (y-k )^2 = r^2` centred at `A = (h, k)` and
radius `r`.

Let `P (x, y)`, the coordinates of P can be expressed as `x = h + r cos theta` and `y= h + r sin theta`

These equations represent the coordinates of any point on the circle in terms of the parameter `theta`.






Position of a Point with Respect to a Circle

A point `(x_1, y_1)` lies outside, on or inside a circle `S = x ^2 + y^2 + 2gx + 2fy + c = 0`

According as `S_1 > ,= `or `< 0` respectively

where, `S_ 1 =x_1^2 + y_1^2 + 2 gx_1 + 2fy_1 + c`

Intersection of a Line and a Circle :

Consider the circle `x ^2 + y^2 = r^2` and the line `y = mx +c`.

(i) If `r > | c/(sqrt(1+m^2))| ` then the line intersects the circle at two distinct points.

(ii) If `r= | c/(sqrt (1+m^2)) |` then the line intersects the circle at only one point. It is called a
tangent.

(iii) If `r < |c/(sqrt (1+m^2))|` then the line does not intersect the circle.

Relative position of circles

`(i)text( Circles do not intersect )`

`C_1C_2>r_1+r_2`

Four common tangent can be drawn- two direct `& ` two transverse

`(ii)text(Circles touch each other externally )`

`C_1C_2=r_1+r_2`

Three common tangents can be drawn .

`(iii)text(Circles intersect in two points)`

`|r_1- r_2| < C_1C_2 < r_1+r_2`

Two common tangents can be drawn

`(iv)text(Circles touch each other internally)`

`C_1C_2=|r_1-r_2|`

Only one common tangent can be drawn

`(v)text(One circle lie completely inside other )`

`0<= C_1C_2 < |r_1-r_2|`

No common tangent can be drawn.

 
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