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

What must I remember about Tangents to a circle from an external point?

• From any external point P(x1, y1) lying outside a circle (S1 > 0), exactly TWO real tangents can be drawn to the circle.
• Length of Tangent: The distance from external point P(x1, y1) to either point of tangency on circle x² + y² + 2gx + 2fy + c = 0 is L = √S1 = √(x1² + y1² + 2gx1 + 2fy1 + c).
• Pair of Tangents: The combined joint equation of the pair of tangents drawn from P(x1, y1) to circle S = 0 is S*S1 = T².
• Chord of Contact: The straight line joining the two points of tangency of tangents drawn from external point P(x1, y1) has equation T = 0 (x*x1 + y*y1 + g(x + x1) + f(y + y1) + c = 0).
• Angle θ between pair of tangents: tan(θ/2) = r / √S1, where r is radius of circle and √S1 is length of tangent.

Test yourself (1 minute)

1. Method 1 vs Method 2 for finding individual tangents

   Method 1 uses point-gradient line y - y1 = m(x - x1) and equates perpendicular distance from center to radius r to solve for the two values of m; Method 2 solves the quadratic joint equation S*S1 = T² into linear factors.

2. Vertical tangent awareness

   If the quadratic in m yields only one finite solution, the second tangent is vertical (equation x = x1 with infinite gradient).

3. Director Circle

   The locus of the point of intersection of two perpendicular tangents to circle x² + y² = a² is the concentric circle x² + y² = 2a² (radius = a√2).

Practise Tangents to a circle from an external point questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 19

What should I study first?

What comes next?

Next in the syllabus

Where does Tangents to a circle from an external point fit in the syllabus?

Year 1 · Term 3