Tangents to a circle from an external point — A/L Combined Mathematics

From an external point P(x₁, y₁) lying outside a circle S ≡ x² + y² + 2gx + 2fy + c = 0, exactly two real tangents can be drawn to the circle. The length of each tangent from P to the point of contact is given by L = √S₁ = √(x₁² + y₁² + 2gx₁ + 2fy₁ + c). The combined equation of the pair of tangents drawn from P(x₁, y₁) to the circle is given by SS₁ = T², where T ≡ xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c. The line joining the two points of contact is called the chord of contact, and its equation is simply T = 0.

Check before you answer (A/L Pure Maths Paper 1)

1. Tangent Length Verification: Verifying that S₁ > 0 confirming P lies strictly outside the circle before computing tangent length L = √S₁.
2. Pair of Tangents Formula: Correctly formulating and expanding SS₁ = T² to find the separate gradients of both tangent lines.
3. Chord of Contact vs Tangent: Distinguishing clearly between the equation of a tangent at a point on the circle (T = 0, where S₁ = 0) and the chord of contact from an external point (T = 0, where S₁ > 0).