(A) Point form

Let `P(x_1 , y_1)` be the point of circle `x^2 + y^2 = a^2`

Since `C` the centre of the circle has co-ordinates `(0, 0)` therefore

slope of `CP = (y_1 - 0)/(x_1 - 0) = (y_1)/(x_1)`

lf `m` is the slope of the tangent at `P` then

`m (y_1//x_1) =-1` (`:.` tangent is `bot` `CP`)

or `m = -x_1//y_1`

The equation of the tangent at `P(x_1, y_1)` is

`y-y_1=-x_1/y_1(x-x_1)`

or `yy_1 - y_1^2 = -x x_1 + x_1^2` or `x x_1 + yy_1 = x_1^2 + y_1^2 = a^2`

[`:. (x_1, y_1)` lies on the circle `x^2 + y^2 = a^2` `:. x_1^2 + y_1^2 = a^2`}

Hence the equation of the tangent at `(x_1, y_1)` is

` x x_1+yy_1 = a^2` or

`T = 0` ( Here `T= x.x_1+yy_1 - a^2` )

Funda 1 :

Equation of tangents drawn to any second degree circle at `P (x_1 y_1)` on it can be obtained by replacing.

`x^2 -> x x_1 ; y^2 -> y y_1 ; 2x -> x + x_1 ; 2y -> y + y_1 ; 2x y -> x y_1 + y x_1`

Funda 2 :

Point of Tangency:

for `P` : either solve tangent and normal to get `P` or compare the equation of tangent at `(x_1 ,y_1)` with the given tangent to get point of tangency

Let the equation of circle is `x^2 + y^2 = a^2` slope of tangent is m then, equation of tangent will be `y = mx + c` when `c` is constant. Again if `y= mx +c` is tangent for circle then apply the condition of tangency

`|c/sqrt(1 + m^2)| = a` or `c= +-sqrt(1 + m^2)`

Equation of tangent

`y = mx +- a sqrt(1 + m^2)`

Let the equation of circle is `x^2 + y^2 = a^2`

Then equation of tangent for point `(x_1, y_1)` on circle is

`x x_1+ yy_1=a^2`

For parametric equation `x_1 = a cos theta` and `y_1 = a sin theta`

`:. x(a cos theta) +y(a sin theta) = a^2`

`x cos theta + y sin theta = a`

Let the equation of circle is `x^2 + y^2 = a^2`

Let `P(x_1, y_1)` is any external point for circle then equation of tangent will be `(y-y_1) = m(x-x_1)`

Form apply the condition of tangency get the two values of `m`

Note:

(i) For a unique value of `m` there will be `2` tangent which are parallel to each other.

(ii) From an external point `2` tangents can be drawn to the circle which are equal in length and are equally inclined to the line joining the point and the centre of the circle.

Definition : The normal to a circle at a point is defined as the straight line passing through the point and perpendicular
to the tangent at that point.

`text(Clearly every normal passes through the centre of the circle. )`

The equation of the normal to the circle `x^2+ y^2+ 2gx + 2fy+c =0`

at any point `(x_1 , y_1)` lying on the circle is `(y_1 + f)/(x_1 + g) = (y - y_1)/(x - x_1)`

In particular, equation of the Normal to the circle

`x^2 + y^2 = a^2` at `(x_1, y_1)` is `y/x = (y_1)/(x_1)`

Let the equation of circle is `S = x^2 + y^2 + 2gx + 2fy + c = 0` and length of tangent is `L` then,

`L_T^2 + R^2 = d^2`

`L_T^2 =d^2 - R^2`

`L_T^2= (x_1 +g)^2 + (y_1 +f)^2 - (g^2 +f^2 - c)`

`L_T^2= x_1^2 + y_1^2 + 2gx_1 +2fy_1 + c`

`:. L _T= sqrt(x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c )= sqrt(S_1)`

NOTE : All these formulae are applicable when coeff. of `x^2` & `y^2` is unity

`PT^2 = S_1`

Square of the length of the tangent from the point `P` is called power of the point `P` w.r.t a given circle i.e. `PT^2 = S_1`

Power of a point remains constant w.r.t a circle

`PA * PB = (PT)^2`

Analytical proof :

`(x-x_1)/(cos theta) = (Y-Y_I)/(sin theta) = r` ;substituting `x = x_1 + r costheta` and `y = y_1 + r sintheta` in `x^2 + y^2 = a^2`,

we get,

`r^2 + 2r (x_1 cos theta + y_1 sin theta) + x_1^2 + y_1^2 - a^2 = 0`

`r_1r_2 = x_1^2 + y_1^2 - a^2 =` constant `= (PT)^2`

Note: Power of a point is+ ve/ 0 (zero) I-ve according as point `P` lies outside/ on/ inside the circle.

(i) Area of Quad PAOB `= 2 triangle POA`

` = 2. 1/2 RL= R L`

(ii) Find `AB` i.e length of chord of contact

`AB = 2L sin theta` where `tan theta = R/L`

`=(2RL)/(sqrt(R^2 + L^2)`

(ill) Area of `trianglePAB` (`triangle`formed by pair of Tangent & corresponding C.O.C.)

`triangle PAB = 1/2 AB xx PD`

`= 1/2 (2L sin theta)(L cos theta)`

`=L^2 sin theta cos theta`

`= (RL^3)/(R^2 + L^2)`

(iv) Angle `2 theta` between the pair of Tangent

`tan 2theta = (2 tan theta)/(1 - tan^2 theta) = (2RL^2)/(L(L^2 - R^2))`

`2 theta = tan^(-1) ((2RL)/(L^2 - R^2))`

Director circle is a name given to a special locus. Locus of a point `P` which moves in such a way such that the pair of tangents drawn from `P` to a given circle makes an angle of `90-`

OR

Locus of the point of intersection of two mutually perpendicular tangents drawn to a given circle is called the director circle of the given circle.


The equation of the director circle of the circle `x^2 + y^2 = a^2` is

`x^2 + y^2 = 2a^2`

Proof:

The equation of any tangent to the circle `x^2 + y^2 = a^2` is

`y = mx + a sqrt(1 + m^2)`......................`(i)`

Let `P (h, k)` be the point of intersection of tangents, then `P (h, k)` lies on `(i) `

`:. k = mh + a sqrt(1 + m^2)`

or `(k - mh)^2 = a^2 (1 + m^2)`

or `m^2 (h^2 - a^2) - 2mkh + k^2 - a^2 = 0`

This is quadratic equation in `m` ,let two roots are `m_1` and `m_2`.
But tangents are perpendiculars, then

`m_1 m_2 = -1`

`=> (k^2 - a^2)/(h^2 - a^2) = -1`

or `k^2 - a^2 = -h^2 + a^2`

or ` h^2 + k^2 = 2a^2`

Hence locus of `P (h, k)` is `x^2 + y^2 = 2a^2`

`(a) text( Direct Common Tangents:)`

Direct common tangent is a tangent touching two circles at different points and not intersecting the line of centres between the centres as shown in figure-1.

`(b) text( Transverse Common Tangents:)`

Transverse common tangent is a tangent touching two cirlces at different points and intersecting the line of centres between the centres as shown in figure-1.

`(c)text(Relative Position of Circles And Number of Common
Tangents:)`

`(i) quad C_1C_2 > r_1+r_2` Circles not intersect.
Four common tangent can be drawn-two direct & two transverse.



`(ii) quad C_1C_2=r_1+r_2` Circles touch each other externally
Three common tangents can be drawn



`(iii) quad |r_1-r_2|< C_1C_2< r_1+r_2` Circles intersect in two points.
Two common tangents can be drawn



`(iv) quad C_1C_2=|r_1-r_2|` Circles touch each other internally
Only one common tangent can be drawn




`(v) quad 0 le C_1C_2<|r_1-r_2|` One circle lie completely inside other
No common tangent can be drawn