Ellipse – Tangent Tangent to the Ellipse: Let S = 0 be an ellipse and P be a point on the ellipse. Let Q be any other point on the ellipse. If the secant line PQ approaches to the same limiting position as Q moves along the curve and approaches to P from either side, Read more about Ellipse – Tangent[…]

Hyperbola – Tangent and Normal Tangent to the Hyperbola: Let S = 0 be a hyperbola and P be a point on the hyperbola Let Q be any other point on the hyperbola. If the secant line PQ approaches to the same limiting position as Q moves along the curve and approaches to P from Read more about Hyperbola – Tangent and Normal[…]

Hyperbola – Theorems Theorem Statement: The equation of the chord joining the two points A(x₁, y₁), B(x₂, y₂) on the hyperbola S = 0 is S₁ + S₂ = S₁₂ Theorem: The difference of the focal distance of any point on the hyperbola is constant i.e., if P is appointed on the hyperbola with foci Read more about Hyperbola – Theorems[…]

Parabola – Two parabolas intersect at an angle Prove that the two parabolas y² = 4ax and x² = 4by intersect (other then the origin) at an angle of . Solution: Without loss of generality we can assume that a > 0 and b > 0 Let P (x, y) be the point of intersection Read more about Parabola – Two parabolas intersect at an angle[…]

Parabola – Tangent and Chord joining Two Points The equation of the chord joining two points A (x₁, y₁), B (x₂, y₂) on the parabola S = 0 is S₁ + S₂ = S₁₂. Proof: Let S = y² – 4ax = 0 be the given parabola Since A and B are two points on Read more about Parabola – Tangent and Chord joining Two Points[…]

Parabola Theorem: Three normal can be drawn from a point (x₁, y₁) to the parabola y² = 4ax. If the normal at t₁ and t₂ to the parabola y² = 4ax meet on the parabola, then t₁t₂ = 2. Proof: Let the normal at t₁ and t₂ meet at t₃ on the parabola The equation Read more about Parabola[…]

Length of a Vector and Division Formulas Length of a vector in term of in components If r = xi + yj + zk then |r| = . Proof: Let then P = (x, y, z) Now |r| = OP = , OP = . Examples: r = 2i + 3j – 6k then find Read more about Length of a Vector and Division Formulas[…]

Periodicity and Extreme Value – Problems Find the periods for the given functions. 1. Cos (3x + 5) + 7 Solution: Given that Cos (3x + 5) + 7 Period of cosx = 2π Period = \(\frac{period\ of\ \cos x}{[coefficient\ of\ x]}=\frac{2\pi }{3}\) 2. Tan5x Solution: Given that Tan5x f(x) = tan5x Period of tanx Read more about Periodicity and Extreme Value – Problems[…]

Angle between two Circles Statement: If θ is the angle between the circles x² + y² + 2gx + 2fy + c = 0, x² + y² + 2g’x + 2f’y + c = 0 then . Proof: Let C₁, C₂ be the centre S and r₁, r₂ be the radii of the circles S = Read more about Angle between two Circles[…]

Variance and Standard Deviation – Problems 1. Find the variance for the discrete data given below 6, 7, 10, 12, 13, 4, 8, 12. Solution: 6, 7, 10, 12, 13, 4, 8, 12 Mean (x̄) = = 9, xᵢ xᵢ – x̄ (xᵢ – x̄)² 6 6 – 9 = -3 9 7 7 – Read more about Variance and Standard Deviation – Problems[…]