FORMULA LIBRARY : CIRCLES

CIRCLES

— JEE MATH APEX —

JEE MATH APEX

1. Standard Forms of Circle

• Centre-radius form: (x − h)² + (y − k)² = r²
• Centre: (h, k), Radius: r
• General form: x² + y² + 2gx + 2fy + c = 0
• Centre: (−g, −f), Radius: √(g² + f² − c)
• Circle with centre at origin: x² + y² = r²

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2. Diameter Form

• Circle with diameter endpoints (x₁,y₁) and (x₂,y₂):
  (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0
• Centre: ((x₁+x₂)/2, (y₁+y₂)/2)
• Radius: (1/2)√[(x₁−x₂)² + (y₁−y₂)²]

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3. Parametric Form

• For circle x² + y² = r²: x = r cos θ, y = r sin θ
• For circle (x−h)² + (y−k)² = r²: x = h + r cos θ, y = k + r sin θ
• Parametric point: (r cos θ, r sin θ)
• Any point on circle can be represented parametrically

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4. Position of a Point

• For circle S = x² + y² + 2gx + 2fy + c = 0 and point (x₁,y₁):
• If S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c
• S₁ < 0: Point inside circle
• S₁ = 0: Point on circle
• S₁ > 0: Point outside circle

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5. Tangent to Circle

• Tangent at point (x₁,y₁) on circle x² + y² = r²: xx₁ + yy₁ = r²
• Tangent to x² + y² + 2gx + 2fy + c = 0 at (x₁,y₁): xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c = 0
• Tangent with slope m to x² + y² = r²: y = mx ± r√(1+m²)
• Condition for tangency of y = mx + c: c = ± r√(1+m²)

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6. Normal to Circle

• Normal at (x₁,y₁) on circle x² + y² = r²: y = (y₁/x₁)x (passes through centre)
• Normal to x² + y² + 2gx + 2fy + c = 0 at (x₁,y₁): (y−y₁)(x₁+g) = (x−x₁)(y₁+f)
• Normal always passes through centre of circle

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7. Length of Tangent

• Length of tangent from (x₁,y₁) to circle x²+y²+2gx+2fy+c=0:
  L = √(x₁² + y₁² + 2gx₁ + 2fy₁ + c) = √S₁
• Length of tangent from origin: √c
• Pair of tangents from (x₁,y₁): SS₁ = T²
  where T = xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c

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8. Chord of Contact

• Chord of contact from (x₁,y₁) to circle x²+y²+2gx+2fy+c=0:
  xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c = 0
• For circle x² + y² = r²: xx₁ + yy₁ = r²
• Chord of contact is perpendicular to line joining centre and external point

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9. Chord with Midpoint

• Chord with midpoint (x₁,y₁) of circle x²+y²=r²: xx₁ + yy₁ = x₁² + y₁²
• For general circle: T = S₁
• Where T = xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c
• And S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c

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10. Director Circle

• Director circle of x² + y² = r²: x² + y² = 2r²
• Director circle of x² + y² + 2gx + 2fy + c = 0:
  (x+g)² + (y+f)² = 2(g² + f² − c)
• Locus of point of intersection of perpendicular tangents

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11. Family of Circles

• Family through intersection of S₁=0 and S₂=0: S₁ + λS₂ = 0
• Family through intersection of circle S=0 and line L=0: S + λL = 0
• Concentric circles: Same centre, different radii: x² + y² + 2gx + 2fy + λ = 0

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12. Common Tangents

• Direct common tangent: √(d² − (r₁−r₂)²)
• Transverse common tangent: √(d² − (r₁+r₂)²)
• Where d = distance between centres, r₁ and r₂ are radii
• Number of common tangents depends on relative position of circles

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13. Orthogonal Circles

• Two circles x²+y²+2g₁x+2f₁y+c₁=0 and x²+y²+2g₂x+2f₂y+c₂=0 are orthogonal if:
  2(g₁g₂ + f₁f₂) = c₁ + c₂
• Orthogonal circles intersect at right angles
• Radii are perpendicular at point of intersection

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14. Radical Axis

• Radical axis of two circles S₁=0 and S₂=0: S₁ − S₂ = 0
• Radical axis is a straight line perpendicular to line joining centres
• Radical centre: Intersection of radical axes of three circles (taken in pairs)
• Radical centre is equidistant from all three circles

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15. Power of a Point

• Power of point P(x₁,y₁) w.r.t. circle S=0: S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c
• Power = (Length of tangent)² = PA × PB (for secant through P)
• If P is inside: Power is negative
• If P is outside: Power is positive
• If P is on circle: Power is zero