Position of a straight line relative to a circle — A/L Combined Mathematics

What must I remember about Position of a straight line relative to a circle?

• Line-Circle geometric relationship: Let circle have center C and radius r, and let d be the perpendicular distance from center C to straight line L.
• Three relative positions: 1) Secant line intersecting circle at two distinct real points: d < r; 2) Tangent line touching circle at exactly one point: d = r; 3) Non-intersecting line lying entirely outside circle: d > r.
• Algebraic intersection condition: Substitute y = mx + c into x² + y² = a² to obtain quadratic (1 + m²)x² + 2mc*x + (c² - a²) = 0; discriminant Δ = 4a²(1 + m²) - 4c² = 4[a²(1 + m²) - c²].
• Condition of Tangency: Straight line y = mx + c is tangent to circle x² + y² = a² if and only if c² = a²(1 + m²), yielding tangent equations y = mx ± a√(1 + m²).
• Length of chord intercepted by circle: If line intercepts chord AB, its length is 2√(r² - d²), where d is the perpendicular distance from center to the line.

Test yourself (1 minute)

1. Equation of tangent at point P(x1, y1) on circle S = 0

   For x² + y² = a², tangent is x*x1 + y*y1 = a²; for general circle x² + y² + 2gx + 2fy + c = 0, tangent is x*x1 + y*y1 + g(x + x1) + f(y + y1) + c = 0 (T = 0).

2. Equation of Normal at P(x1, y1)

   Normal passes through center (-g, -f); for x² + y² = a², normal is x*y1 - y*x1 = 0 (y = (y1/x1)x).

3. Midpoint of chord

   The chord of circle S = 0 whose midpoint is (x1, y1) has equation T = S1.

Practise Position of a straight line relative to a circle questions today

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

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Year 1 · Term 3