Complex Numbers Complex Number: If a, b are two real numbers, then a number of the form a + ib is called a complex number. Examples: 7 + 3i -1 + i 3 + 2i 0 + 3i etc. are complex numbers. Real and Imaginary Parts of a Complex Number: If z = a + Read more about Complex Numbers[…]
Circle – General Equation Definition: A circle is the locus of a point which moves in a plane such that its distance from a fixed point in plane is always a constant. The fixed point is called the center and the constant distance is called the radius of the circle. Equation of Circle (h, k) and Read more about Circle – General Equation[…]
Properties of Tangent – Parabola I (iii) Length of Tangent Between the Point of Contact P(t) and the point where it meets the Directrix Q subtends right angle at focus: Equation of the tangent to the parabola y² = 4ax at point P(t) is ty = x + at². It meets x = – a Read more about Properties of Tangent – Parabola I[…]
Properties of Tangents – Parabola (i) Point of Intersection of Tangent at any Two Points on the Parabola. Let the equation of the parabola be y² = 4ax. The two points on the parabola are P ≡ (at₁², 2at₁) and Q ≡ (at₂², 2at₂). The equation of the tangents at P and Q are, respectively, Read more about Properties of Tangents – Parabola[…]
Hyperbola – Position of a Point with respect to a Hyperbola and Conjugate Position of a Point with respect to a Hyperbola: The quantity is positive zero (or) negative according as the point (x₁, y₁) lies within, upon or outside the curve, respectively. If (x₁, y₁) lies on the hyperbola. Then . = y₁². Read more about Hyperbola – Position of a Point with respect to a Hyperbola and Conjugate[…]
Chord of Contact – Ellipse Let PQ and PR be the tangent to the ellipse drawn from any external point P (h, k). Then QR is called the chord of contact of the ellipse. Let Q ≡ (x₁, y₁) and R = (x₂, y₂) Therefore, the equation of tangents PQ and PR respectively are … Read more about Chord of Contact – Ellipse[…]
General Equation of Second Degree The most general from of a quadratic equation in x and y is ax² + 2hxy + by² + 2gx + 2fy + c = 0. Since it is an equation in x and y, it must represent the equation of a locus in a plane. It may represent a Read more about General Equation of Second Degree[…]
Equation of Tangent – Ellipse Equation of tangent to the ellipse at point (x₁, y₁). , Differentiation with respect to x , , , dy/dx at point (x₁, y₁) , Hence the equation of the tangent is , , But (x₁, y₁) lies on the ellipse ∴ , , (or) T = 0 Where, . Read more about Equation of Tangent – Ellipse[…]
Equation of Pair of Tangents from Point – Parabola Let T (h, k) be nay point on the pair of tangents PQ or PR draw P (x₁, y₁) to the parabola y² = 4ax. The equation of PT is y – y₁ = (k – y₁)/ (h – x₁) (x – x₁) (or) y = Read more about Equation of Pair of Tangents from Point – Parabola[…]
Geometry with Complex Numbers Section Formula: If P(z) divides the line segment joining A(z₁) and B(z₂) internally in the ratio m:n, then . , If the division is external, then . Explanation: Let z₁ = x₁ + iy₁, z₂ = x₂ + iy₂. Then, A Ξ (x₁, y₁) and B Ξ (x₂, y₂). Let z Read more about Geometry with Complex Numbers[…]