Circle ⚫Introduction ⚪ A circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane is always constant. A fixed point is called the centre of the circle and the distance is called the radius of the circle. General equation of conic : ax 2 + by 2 + 2hxy + 2gx + 2fy + c = 0 Condition for being Circle : i) h = 0 ii) abc + 2fgh -af 2 - bg 2 -ch 2 ≠ 0 iii) a= b = 1 ⚪ Equation of circle : x 2 + y 2 + 2gx + 2fy + c = 0 center of circle : (-g , -f) Radius = (g 2 + f 2 -c) 1/2 If g 2 + f 2 -c > 0 , then circle will be real circle If g 2 + f 2 -c = 0 , then circle will be point circle If g 2 + f 2 -c < 0 , then circle will be imaginary circle ⚫ Different forms of equations of Circle ⚪ Central form Let C be the centre of the circle and its coordinates ( h, k ) . Let the radius of the circle be a . Let ( x,y ) be any point on...

Inverse Trigonometry ⚫Introduction ⚪ The inverse of a function f: A → B exists if f is one -one onto i.e. a bijection and is given by f(x) = y ⇒ f -1 (y) =x . ⚪ Consider the sine function with domain R and range [-1,1]. Clearly this function is not a bijection and so it is not invertible. If we restrict the domain of it in such a way that it becomes one-one , then it would become invertible. If we consider sine as a function with domain [-π/2, π/2 ] and co-domain [-1,1] , then it is a bijection and therefore , invertible. The inverse of sine function is defined as sin -1 x = θ where θ → [-π/2, π/2 ] and x → [-1,1] ⚫ Domain and range of trigonometry function ⚫ Basic Formula of Inverse Trigonometry : sin -1 (-x) = -sin -1 x cos -1 (-x) = π -cos -1 x tan -1 (-x) = -tan -1 x cot -1 (-x) = π -cot -1 x sec -1 (-x) = π -sec -1 x cosec -1 (-x) = -cosec -...

Measurement of Angle ,Trigonometric Ratios and equation ⚫Relation Between Degree and Radians ⚫ Relation Between Trigonometrical Ratios and Right Angled Triangle Side ⚫ Relation Between Trigonometrical Ratios sinθ.cosecθ = 1 tanθ.cotθ = 1 cosθ.secθ = 1 tanθ = sinθ/cosθ cotθ = cosθ/sinθ ⚫ Fundamental trigonometric Identities : sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = cosec 2 θ ⚫ Trigonometrical ratios for various angles ⚫ Formula For the trigonometric ratios of sum and difference of two angles sin(A+B) = sinA cosB + cosA sinB sin(A-B) = sinA cosB - cosA sinB cos(A+B) = cosA cosB - sinA sinB cos(A-B) = cosA cosB + sinA sinB sin(A+B).sin(A-B) = sin 2 A - sin 2 B = cos 2 B - cos 2 A cos(A+B).cos(A-B) = cos 2 A - sin 2 B = cos 2 B - sin 2 A ⚫ Formulae to tran...

Properties of triangle ⚫ Relation between sides and angles ⚪ The law of sines or sine rule : The sides of the triangle are proportional to the sines of the angles opposite to them ⚫ The law of cosine or cosine Law : ⚪ Projection Formula : a = b.cosC + c.cosB b = c.cosA + a.cosC c = a.cosB + b.cosA ⚫ Law of tangents : ⚫ Area of triangle ⚫ Half angle Formulae : Practice set 1 Practice set 2

Complex Number ⚫What is Complex number ? ⚪ Combination form of real and imaginary number z = a + ib where i(iota) = √ -1 ⚫ Integer power of i (√ -1 ) (a) i 4n = 1 (b) i 4n+1 = i (c) i 4n+2 = -1 (d) i 4n+3 = -i ⚫ Equality of two complex numbers ⚪ Two complex number z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 are said to be equal if and only if their real and imaginary parts are separately equal. z 1 = z 2 ⇔ x 1 + iy 1 = x 2 + iy 2 ⇔ x 1 = x 2 and y 1 = y 2 ⚫ Conjugate of a complex number : ⚪ For any complex number z = a + ib , then its conjugate is defined as z = a - ib ⚪ Geometrically , the conjugate of z is the reflection or point image of z in the real axis. ⚫ Properties of conjugate : ⚪   z̿ = z ⚪ ⚪ ⚪ ⚪ ⚪ ⚪ ⚪ z + z̅ = 2Re(z) = purely real ⚪ z - z̅ = ...