GUJCET Mathematics – Circles
Topic: Circles
Exam: GUJCET (Gujarat Common Entrance Test)
Level: 12th‑grade / JEE‑Mains level
Goal of these notes:
- Give a simple description of the subject.
- Highlight the key concepts and formulae that appear in GUJCET.
- Provide a worked example that demonstrates how to solve a typical problem.
- Offer a quick revision summary to cement the main ideas.
1. Simple Explanation
| Concept | What it means | Why it matters in GUJCET |
|---|---|---|
| Circle | Set of all points in the plane that are at a fixed distance (r) (radius) from a fixed point (C(h,k)) (centre). | Many geometry and coordinate‑geometry problems involve circles. |
| Radius | Distance from centre to any point on the circle. | Appears in area, circumference, and equations. |
| Diameter | Line segment through centre connecting two points on the circle. Length (=2r). | Used to check whether chords are diameters and to find radius from a chord. |
| Chord | Any line segment whose endpoints lie on the circle. | Central angles, inscribed angles, and chord lengths are common problems. |
| Tangent | Line that touches the circle at exactly one point. | Tangent‑secant theorem, length of tangent segment from an external point, etc. |
| Secant | Line that intersects the circle at two points. | Secant‑tangent theorem, power of a point. |
| Arc | Portion of the circle’s circumference bounded by two points. | Arc length, sector area, inscribed angles. |
2. Key Points & Formulae
2.1. Equations of a Circle
| Form | Equation | Conditions | Notes |
|---|---|---|---|
| Standard form | ((x-h)^2+(y-k)^2=r^2) | (r>0) | (C(h,k)) is centre, (r) radius. |
| General form | (x^2+y^2+Dx+Ey+F=0) | (D^2+E^2-4F>0) | Can be converted to standard form by completing the square. |
| Parametric form | (x = h+r\cos\theta,; y = k+r\sin\theta) | (0\le\theta<2\pi) | Useful for arc length and area of sector. |
Completing the square
[
x^2+y^2+Dx+Ey+F=0 ;; \Longrightarrow ;; (x+\tfrac{D}{2})^2+(y+\tfrac{E}{2})^2 = \tfrac{D^2+E^2-4F}{4}
]
(r = \sqrt{\tfrac{D^2+E^2-4F}{4}}).
2.2. Distance & Power of a Point
| Concept | Formula | Interpretation |
|---|---|---|
| Distance from point to centre | (d = \sqrt{(x-x_0)^2+(y-y_0)^2}) | Used to decide if a point lies inside, on, or outside the circle. |
| Power of a point | (P = d^2 - r^2) | If (P>0): point outside → secant‑tangent theorem. If (P=0): point on circle. If (P<0): point inside. |
| Secant–tangent theorem | For external point (P): (PA \cdot PB = PT^2) (where (A,B) are intersection points of a secant and (T) is point of tangency). | Handy for finding lengths of tangents or chord segments. |
2.3. Tangent Properties
| Property | Formula |
|---|---|
| Slope of tangent (for circle ((x-h)^2+(y-k)^2=r^2)) | ((x_1-h)(x-x_1) + (y_1-k)(y-y_1)=0) at point ((x_1,y_1)) on circle. |
| Equation of tangent (at point ((x_1,y_1))) | ((x_1-h)(x-h)+(y_1-k)(y-k)=r^2). |
| Length of tangent from external point ((x_0,y_0)) | (PT = \sqrt{(x_0-h)^2+(y_0-k)^2 - r^2}). |
2.4. Chords & Angles
| Concept | Formula / Theorem |
|---|---|
| Chord length (distance between two points ((x_1,y_1)) and ((x_2,y_2)) on circle) | (L = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}). |
| Chord perpendicular to radius | If a line through centre is perpendicular to a chord, the chord is |