KCET Mathematics – Study Notes on Circles
Purpose:
These notes give a concise, self‑contained refresher on all the circle concepts you’ll need for the KCET exam.
They are organized into four parts:
- Simple explanation & basic definitions
- Key points (theorems, formulas, properties)
- A worked example that ties the concepts together
- A quick revision summary
Tip: For every theorem or formula, try to draw a diagram first. Visualising the geometry often turns a confusing statement into an obvious fact.
1. Simple Explanation
A circle is the set of all points in a plane that are equidistant from a fixed point called the centre.
If the common distance is (r) (the radius), then every point (P(x,y)) on the circle satisfies
[ (PC)^2 = r^2 ]
where (C) is the centre of the circle.
Basic Terminology
- Centre – the fixed point (C).
- Radius – line segment (CR) from centre to any point (R) on the circle.
- Diameter – any chord that passes through the centre; its length is (2r).
- Chord – a line segment whose endpoints lie on the circle.
- Tangent – a line that touches the circle at exactly one point.
- Secant – a line that cuts the circle at two points.
- Arc – part of the circumference between two points.
- Sector – region bounded by two radii and the arc between them.
- Segment – region bounded by a chord and the arc it subtends.
2. Key Points
| Category | Concept | Formula / Statement | Notes |
|---|---|---|---|
| Basic Geometry | Circumference | (C = 2\pi r) | |
| Area | (A = \pi r^2) | ||
| Diameter | (d = 2r) | ||
| Theorem – Tangent & Chord | Tangent–Chord Angle Theorem | Angle between a tangent and a chord through the point of contact = angle in the alternate segment (subtended by the same chord). | Useful for proving cyclic quadrilaterals. |
| Tangent–Tangent Angle Theorem | Angle between two tangents = difference of their intercepted arcs. | ||
| Tangent–Secant Angle Theorem | Angle between a tangent and a secant = half the difference of the intercepted arcs. | ||
| Perpendicular from centre to chord | The line joining centre to the midpoint of a chord is perpendicular to the chord. | ||
| Equal chords subtend equal angles | If two chords are equal in length, the angles subtended by them at the centre are equal. | ||
| Power of a Point | For point (P) outside the circle: (PA \cdot PB = PT^2) (where (A,B) are intersection points of a secant, (T) a point of tangency). | ||
| Coordinate Geometry | General equation | ((x-a)^2 + (y-b)^2 = r^2) where ((a,b)) is centre. | |
| Equation of tangent at ((x_1,y_1)) | ((x_1-a)(x-a)+(y_1-b)(y-b)=r^2) (or simply ((x_1-a)(x-a)+(y_1-b)(y-b)=r^2)). | Equivalent to ((x_1-a)x + (y_1-b)y = a^2+b^2-r^2). | |
| Distance from point to centre | (\sqrt{(x-a)^2+(y-b)^2}) | Compare with (r) to decide inside/outside. | |
| Intersection of two circles | Solve system of their equations; discriminant tells number of intersections. | ||
| Chord length from centre distance | If chord is at distance (d) from centre, length (= 2\sqrt{r^2-d^2}). | ||
| Special Circles | Circumcircle of triangle | Circle passing through all three vertices. | Radius (R = \frac{abc}{4\Delta}). |
| Incircle of triangle | Circle tangent to all three sides. | Radius (r = \frac{\Delta}{s}). | |
| Nine‑point circle | Passes through the midpoints of sides, feet of altitudes, midpoints of segments from orthocentre to vertices. | Radius = (R/2). | |
| Trigonometric Relations | Law of sines in a circle | (\frac{a}{\sin A} = 2R) | |
| Power of a point (trig) | (PA \cdot PB = PC^2 - r^2) | ||
| Pythagorean‑like Relation | In right triangle inscribed in circle | Hypotenuse is diameter. |
3. Worked Example
Problem (KCET style)
*A circle has centre (C(2,3)). It passes through the point (A(5,7)) and is tangent to the line (x-2y+5=0). Find the equation of the circle and the length of