Circle — MHT-CET Maths

Formulas (4)

• Centre-Radius and General Forms: Read (−g, −f) and √(g² + f² − c) · Two forms
  $$(x - h)^2 + (y - k)^2 = r^2 \qquad x^2 + y^2 + 2gx + 2fy + c = 0:\ C(-g, -f),\ r = \sqrt{g^2 + f^2 - c}$$
• Diameter Form: (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0 · Diameter form
  $$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \iff x^2 + y^2 - (x_1 + x_2)x - (y_1 + y_2)y + x_1x_2 + y_1y_2 = 0$$
• Parametric Form: x = h + r cos θ, y = k + r sin θ · Parametric circle
  $$x = h + r\cos\theta,\quad y = k + r\sin\theta \qquad (0 \le \theta < 2\pi)$$
• Points of a Family on a Circle: Substitute, Get a Polynomial, Use Vieta · Vieta on the substituted polynomial
  $$m^4 + 2gm^3 + cm^2 + 2fm + 1 = 0 \ \Rightarrow\ m_1 m_2 m_3 m_4 = 1$$

Watch out for (4)

• Reading the centre as (g, f)
  → Centre-Radius and General Forms: Read (−g, −f) and √(g² + f² − c)
• Sign of the constant with negative products
  → Diameter Form: (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0
• Halving the radius with the centre
  → Parametric Form: x = h + r cos θ, y = k + r sin θ
• Product of roots with the wrong sign
  → Points of a Family on a Circle: Substitute, Get a Polynomial, Use Vieta

Formulas (2)

• Concentric Circles: Same Centre, New Radius From Area or From a Point · Concentric family
  $$x^2 + y^2 + 2gx + 2fy + \lambda = 0 \qquad \text{double area: } R^2 = 2r^2$$
• Touching a Line or an Axis: Radius = Distance From the Centre; Contact Point = Foot of the Perpendicular · Tangency of a line
  $$\frac{|ah + bk + c|}{\sqrt{a^2 + b^2}} = r \qquad \text{touches the } X\text{-axis} \iff r = |k|$$

Watch out for (2)

• Doubling the radius for double the area
  → Concentric Circles: Same Centre, New Radius From Area or From a Point
• Using the centre's x-coordinate for tangency to the X-axis
  → Touching a Line or an Axis: Radius = Distance From the Centre; Contact Point = Foot of the Perpendicular

Formulas (4)

• Tangent at a Point on the Circle: T = 0 · Tangent at a point
  $$xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 \qquad x^2 + y^2 = a^2:\ x\cos\theta + y\sin\theta = a$$
• Tangent of a Given Slope, and Whether a Line Touches: Distance From the Centre = Radius · Tangency condition
  $$y = mx \pm a\sqrt{1 + m^2} \qquad \text{general: } \frac{|ah + bk + c|}{\sqrt{a^2 + b^2}} = r$$
• Tangents From an External Point: Length √S₁, the Kite, the Angle Between Them · External point
  $$L = \sqrt{S_1} = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c},\qquad [PAOB] = rL,\qquad \sin\frac{\theta}{2} = \frac{r}{CP}$$
• Loci From Tangent Lengths, and PQ · RS = (2r)² · Tangent-length locus
  $$\frac{\sqrt{S_1}}{\sqrt{S_2}} = \frac{m}{n} \iff n^2 S_1 = m^2 S_2 \qquad PQ \cdot RS = (2r)^2$$

Watch out for (4)

• Forgetting to halve the linear coefficients in T
  → Tangent at a Point on the Circle: T = 0
• Using the slope of the given line instead of the perpendicular one
  → Tangent of a Given Slope, and Whether a Line Touches: Distance From the Centre = Radius
• Halving the kite
  → Tangents From an External Point: Length √S₁, the Kite, the Angle Between Them
• Sign of the x term after cross-multiplying
  → Loci From Tangent Lengths, and PQ · RS = (2r)²

Formulas (3)

• Greatest and Least Distance: d ± r From a Point, and From the Circle to a Line · Extreme distances
  $$\min = |d - r|,\quad \max = d + r \qquad (d = \text{distance from the point, or the line, to the centre})$$
• When a Line Cuts the Circle: Distance From the Centre < r · Line and circle
  $$p = \frac{|ah + bk + c|}{\sqrt{a^2 + b^2}}:\quad p < r \text{ cuts},\ p = r \text{ touches},\ p > r \text{ misses};\qquad \text{chord} = 2\sqrt{r^2 - p^2}$$
• Area Cut Off by a Chord, and the Circumcircle of an Equilateral Triangle · Segment and circumradius
  $$\text{segment} = \frac{r^2}{2}(\theta - \sin\theta) \qquad \text{equilateral: median} = \frac{3R}{2}$$

Watch out for (3)

• Adding r twice for the far end of the diameter
  → Greatest and Least Distance: d ± r From a Point, and From the Circle to a Line
• Counting the tangent cases
  → When a Line Cuts the Circle: Distance From the Centre < r
• Taking R as the median
  → Area Cut Off by a Chord, and the Circumcircle of an Equilateral Triangle

Formulas (2)

• Relative Position From d, r₁ + r₂ and |r₁ − r₂|: How Many Common Tangents · Five positions
  $$d > r_1 + r_2:\ 4 \quad d = r_1 + r_2:\ 3 \quad |r_1 - r_2| < d < r_1 + r_2:\ 2 \quad d = |r_1 - r_2|:\ 1 \quad d < |r_1 - r_2|:\ 0$$
• Touching Circles: d = r₁ + r₂ (External) or d = |r₁ − r₂| (Internal), and the Centre From the Contact Point · Touching
  $$\text{external: } d = r_1 + r_2 \qquad \text{internal: } d = |r_1 - r_2| \qquad \text{centres and contact point are collinear}$$

Watch out for (2)

• Comparing d with r₁ + r₂ only
  → Relative Position From d, r₁ + r₂ and |r₁ − r₂|: How Many Common Tangents
• Squaring once and stopping
  → Touching Circles: d = r₁ + r₂ (External) or d = |r₁ − r₂| (Internal), and the Centre From the Contact Point