Circles — NDA Maths

Formulas (8)

• What a Circle Equation Is · Standard form
  $$(x-h)^2 + (y-k)^2 = r^2$$
• General Form — Centre and Radius by Completing the Square · General form
  $$x^2+y^2+2gx+2fy+c=0 \;\Rightarrow\; \text{centre }(-g,-f),\;\; r=\sqrt{g^2+f^2-c}$$
• Diameter Form — Circle From Two Endpoints · Diameter form
  $$(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$$
• Intercepts a Circle Cuts on the Axes · Axis intercept lengths
  $$\text{x-axis: }2\sqrt{g^2-c}\qquad \text{y-axis: }2\sqrt{f^2-c}$$
• Perpendicular From the Centre Bisects a Chord · Chord length from centre distance
  $$\text{chord} = 2\sqrt{r^2 - d^2}\quad(d=\text{distance from centre to the chord})$$
• Circles That Touch the Axes · Tangency condition
  $$\frac{|ah+bk+c|}{\sqrt{a^2+b^2}} = r$$
• Two Circles — Intersecting, Touching, Separate · Two distinct intersections
  $$|r_1 - r_2| < d < r_1 + r_2$$
• Circle Through the Origin With Given Axis Intercepts · Circle through origin, intercepts a, b
  $$x^2+y^2-ax-by=0,\qquad \text{centre }\left(\tfrac a2,\tfrac b2\right)$$

Watch out for (8)

• Divide by the leading coefficient BEFORE reading g, f, c
  → General Form — Centre and Radius by Completing the Square
• Centre is MINUS g and MINUS f
  → General Form — Centre and Radius by Completing the Square
• The x-factors and y-factors are separate
  → Diameter Form — Circle From Two Endpoints
• Intercept is the GAP between roots, not a single root
  → Intercepts a Circle Cuts on the Axes
• Use the NEGATIVE-reciprocal slope for the perpendicular
  → Perpendicular From the Centre Bisects a Chord
• Touching an axis is |coordinate| = r, not coordinate = r
  → Circles That Touch the Axes
• Both inequalities matter — it's a band, not a single bound
  → Two Circles — Intersecting, Touching, Separate
• Through the origin forces the constant term to vanish
  → Circle Through the Origin With Given Axis Intercepts

Formulas (7)

• Building a Circle From Diameter Endpoints · Circle on a diameter
  $$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$$
• Circle Through Three Points — the General-Equation System · Unknown-coefficient circle
  $$x^2+y^2+Dx+Ey+F=0,\quad \text{centre }\left(-\tfrac D2,-\tfrac E2\right)$$
• Extracting Centre and Radius From Three Points · Radius from centre and a point
  $$r^2 = (x_0-h)^2 + (y_0-k)^2$$
• Centre on a Given Line — Perpendicular-Bisector Method · Equidistance condition
  $$(h-x_1)^2+(k-y_1)^2 = (h-x_2)^2+(k-y_2)^2$$
• Concyclicity — Does a Fourth Point Lie on the Circle? · Concyclicity
  $$(x_4,y_4)\text{ on }x^2+y^2+Dx+Ey+F=0 \iff x_4^2+y_4^2+Dx_4+Ey_4+F=0$$
• Family of Circles Through a Chord (the S + λL Trick) · Family through a chord
  $$S + \lambda L = 0$$
• Circumcentre of a Right Triangle — Midpoint of the Hypotenuse · Right-triangle circumcentre
  $$\text{circumcentre} = \text{midpoint of hypotenuse},\quad R=\tfrac12(\text{hypotenuse})$$

Watch out for (6)

• Clear fractions, then match the option's scale
  → Circle Through Three Points — the General-Equation System
• Compute r² and compare squares — skip the root
  → Extracting Centre and Radius From Three Points
• Two through-points give ONE equation, not two
  → Centre on a Given Line — Perpendicular-Bisector Method
• An unknown coordinate gives TWO values — keep both
  → Concyclicity — Does a Fourth Point Lie on the Circle?
• "Chord as diameter" = the new centre sits on the chord line
  → Family of Circles Through a Chord (the S + λL Trick)
• Only works when there IS a right angle
  → Circumcentre of a Right Triangle — Midpoint of the Hypotenuse

Formulas (5)

• Inscribed Angle and the Angle in a Semicircle · Inscribed angle
  $$\angle BAC = \tfrac12\,\angle BOC$$
• Points Where a Circle Touches the Axes · Contact points and PQ
  $$P=(k,0),\;\; Q=(0,k),\;\; PQ=\sqrt2\,|k|$$
• A Square Inscribed in a Circle · Inscribed-square vertices
  $$\left(h\pm\tfrac{r}{\sqrt2},\; k\pm\tfrac{r}{\sqrt2}\right)$$
• Tangent and Normal at a Point of Contact · Opposite end of the diameter
  $$T'=2C-T\quad(C=\text{centre},\;T=\text{contact point})$$
• Areas of the Minor and Major Segments · Minor segment area
  $$A_{\text{minor}} = \tfrac{a^2}{2}\,(\theta - \sin\theta)$$

Watch out for (6)

• Don't forget the supplementary (obtuse) case
  → Inscribed Angle and the Angle in a Semicircle
• A is not pinned to one coordinate
  → Inscribed Angle and the Angle in a Semicircle
• The contact point shares ONE coordinate with the centre
  → Points Where a Circle Touches the Axes
• Inscribed vs circumscribed — offset is r/√2, not r
  → A Square Inscribed in a Circle
• The normal goes through the centre — that's the whole trick
  → Tangent and Normal at a Point of Contact
• Segment = sector − triangle (not sector alone)
  → Areas of the Minor and Major Segments