📚 Study Notes – Circles
Program: DCECE | Subject: Mathematics | Topic: Circles
Circles are one of the most fundamental shapes in geometry. Understanding their properties is essential for solving many problems in algebra, trigonometry, and calculus.
1️⃣ Simple Explanation
A circle is the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center).
- Equation (Cartesian coordinates):
[ (x-h)^2 + (y-k)^2 = r^2 ] where ((h,k)) is the center and (r) is the radius. - Key elements:
- Center (C(h,k))
- Radius (r)
- Diameter (2r) (passes through the center)
- Chord (any line segment whose endpoints lie on the circle)
- Tangent (a line that touches the circle at exactly one point)
- Arc (portion of the circle’s circumference)
2️⃣ Key Points
| Concept | What It Means | Formula / Property |
|---|---|---|
| Circle Equation | Standard form in the plane | ((x-h)^2+(y-k)^2=r^2) |
| Radius & Diameter | Distance from center to any point on the circle | (d = 2r) |
| Chord Length | For a chord at distance (d) from the center | (L = 2\sqrt{r^2-d^2}) |
| Central Angle | Angle subtended by a chord at the center | (\theta = 2\arcsin\left(\frac{L}{2r}\right)) |
| Inscribed Angle | Angle formed by two chords meeting at a point on the circle | (\alpha = \frac{1}{2}\theta) |
| Arc Length | Length of a circular arc | (s = r\theta) (θ in radians) |
| Area of Circle | Total area inside the circle | (A = \pi r^2) |
| Tangent Property | Tangent is perpendicular to radius at the point of contact | ( \text{(radius)} \perp \text{(tangent)}) |
| Secant | Line cutting the circle at two points | Extends beyond the circle |
3️⃣ Worked Example
Problem:
Find the equation of the circle that passes through the points (A(2,4)) and (B(6,0)) and has its center on the line (x = 4).
Solution Steps
-
Let the center be (C(4, k)).
Since the center lies on (x=4), its x‑coordinate is 4; the y‑coordinate is unknown (k). -
Use equal radii to both points
[ r^2 = (2-4)^2 + (4-k)^2 = (6-4)^2 + (0-k)^2 ] Expand: [ (−2)^2 + (4-k)^2 = (2)^2 + (−k)^2 ] [ 4 + (4-k)^2 = 4 + k^2 ] Simplify: [ (4-k)^2 = k^2 ] [ 16 - 8k + k^2 = k^2 ] [ 16 - 8k = 0 \quad\Rightarrow\quad k = 2 ] -
Center found: (C(4,2)).
-
Compute radius (r)
Use point (A): [ r^2 = (2-4)^2 + (4-2)^2 = (-2)^2 + 2^2 = 4 + 4 = 8 ] So (r = \sqrt{8} = 2\sqrt{2}). -
Write the circle equation
[ (x-4)^2 + (y-2)^2 = 8 ]
Answer:
[
\boxed{(x-4)^2 + (y-2)^2 = 8}
]
4️⃣ Quick Revision Summary
- Standard form: ((x-h)^2+(y-k)^2=r^2).
- Radius (r) = distance from center to any point on circle.
- Diameter (d = 2r).
- Chord length (L = 2\sqrt{r^2-d^2}) where (d) is the perpendicular distance from center to chord.
- Central angle ( \theta ) subtends a chord of length (L).
- Inscribed angle is half the central angle that subtends the same arc.
- Arc length ( s = r\theta ) (θ in radians).
- Area ( A = \pi r^2 ).
- Tangent is perpendicular to radius at point of contact.
- Secant passes through the circle cutting it at two points.
Tip: When solving for a circle’s equation, always try to find the center first (often by symmetry or given constraints) and then use the distance formula to get the radius.
Good luck with your studies!