Pair of Tangents – Circles The equation to the pair of tangents to the circle S = 0 from P (x₁, y₁) is S²₁ = S₁₁S. Let the equation of the circle be S ≡ x² + y² + 2gx + 2fy + c = 0 Let a line L = 0 through P (x₁, y₁) Read more about Pair of Tangents – Circles[…]
Polar Equation – Circle Pole and Polar: Let S = 0 be a circle and P be a point. If any line drawn through the point P meets the circle in two points A and B then the point of intersection of the tangent drawn at A and B lie on a line called the Read more about Polar Equation – Circle[…]
Length of the Chord – Circle The length of the chord of the circle S = 0 having P (x₁, y₁) as its midpoint is 2 √|S₁₁|. Proof: Let AB be the chord of the circle S = 0 having P as its midpoint. Now PA = PB and PA. PB = |S₁₁| ⇒ PA² Read more about Length of the Chord – Circle[…]
Circles – Midpoint of Chord The equation of the chord of the circle S = 0 having P (x₁, y₁) as its mid-point is S₁ = S₁₁. Proof: Let the equation of the circle be S = x² + y² + 2gx + 2fy + c = 0. Centre C = (-g, -f) and P Read more about Circles – Midpoint of Chord[…]
Point of Contact and Tangent to the Circle Theorem: The condition that the straight-line lx+ my + n = 0 may touch the circle x² + y² = a² is n² = a² (l² + m²) and the point of contact is . Proof: The given line is lx + my + n = 0 Read more about Point of Contact and Tangent to the Circle[…]
Circles – Power of a Point Let S = 0 be a circle with center C and radius r. let P be a point. Then CP² – r² is called power of P with respect to the circle S = 0. Theorem: The power of a point P (x₁, y₁) with respect to the circle Read more about Circles – Power of a Point[…]
Parametric Equations of a Circle If P (x, y) is a point on the circle with center C (h, k) and radius r, then x = h + r sin θ, y = k = r cos θ where 0 ≤ θ < 2π. Proof: Let θ be the angle made by the line with x Read more about Parametric Equations of a Circle[…]
Mathematical Induction Principle of finite Mathematical Induction: Let {S (n)/ n ϵ N} be a set of statements. Step 1: Let S (n) is given statement, S (1) is true. Step 2: Assume that the statement S (m) is true. Step 3: Assume S (m) is true ⇔ S (m + 1) statement is true. Read more about Mathematical Induction[…]
Equation of a Circle The equation of the circle with center C (h, k) and radius r is (x – h)² + (y – k)² = r². Proof: Above image represents P (x₁, y₁) is locus of the point, r is radius and C (h, k) is center. The distance between the CP is r. Read more about Equation of a Circle[…]
Injection Function and Surjection Functions Injection (or) One – One Function: Definition: A function f: A → B is called an injection. if distinct element f – images in B. an injection is also called a one – one function. f: A → B is an injection ⇔ a₁, a₂ ϵ A and a₁ ≠ a₂ implies Read more about Injection Function and Surjection Functions[…]