Pair of Tangents

Pair of Tangents

Tangents are drawn from P(x₁, y₁) to the circle S = x² + y² + 2gx + 2fy + c = 0.

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

S = x² + y² + 2gx + 2fy + c

S₁ = x₁₂ + y₁₂ + 2gx₁ + 2fy₁ + c

and T = xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + 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(x₁, y₁) and C₂ (x₂, y₂) and radii r₁, r₂ respectively touch each other.

(a) Externally

If |C₁C₂| = r₁ + r₂ and the point of contact is

(b) Internally

If |C₁C₂| = |r₁ – r₂| and the point of contact is