Pair of Tangents

Pair of Tangents

Tangents are drawn from P(x ₁ , y ₁ ) to the circle S = x ² + y ² + 2gx + 2fy + c = 0.

The joint equation of the pair of tangents is SS1 = T2 where

S = x ² + y ² + 2gx + 2fy + c

S ₁ = x ₁₂ + y ₁₂ + 2gx ₁ + 2fy ₁ + c

and T = xx ₁ + yy ₁ + g(x + x ₁ ) + f(y + y ₁ ) + c

Equation of the chord joining two points (acosθ, a sinθ) and (acosφ, asinφ) on the circle x2 + y2 = a2 is

x cos

Note that In case θ → φ it becomes a tangent given by

xcosφ + y sinφ = a

External and Internal contacts of circles

Two circles with centers C1(x ₁ , y ₁ ) and C ₂ (x ₂ , y ₂ ) and radii r ₁ , r ₂ respectively touch each other.

(a) Externally

If |C ₁ C ₂ | = r ₁ + r ₂ and the point of contact is

(b) Internally

If |C ₁ C ₂ | = |r ₁ – r ₂ | and the point of contact is