The equation of a family of circles passing through the intersection of circles x² + y² + 2gx + 2fy + c = 0 and line L = lx + my + n = 0 is x² + y² + 2gx + 2fy + c + λ (lx + my + n) = 0 i.e., Read more about Intersection of line and circle[…]
Radical axis: The radical axis of two circles is the locus of a point which moves in such a way that length of the tangent drawn from it to the two circles are equal. The radical axis of circles S₁ = 0 and S₂ = 0 is given by S₁ – S₂ = 0. Properties Read more about Radical Axis of Two Circles and it’s Properties[…]
Angle of intersection of Two Circles: The angle of intersection of two circles is defined as the angle between the tangents to the two circles at their point of intersection. Let, x² + y² + 2g₁x + 2f₁y + c₁ = 0 x² + y² + 2g₂x + 2f₂y + c₂ = 0. . Orthogonal Read more about Angle between Two Intersecting Circles[…]
Normal: The normal at any point P of the centre is the line which passes through P and is perpendicular to the tangent at P.Equation of Normal: i. The equation of the normal at p(x₁, y₁) of the circle. ⇒ S = 0 ⇒ x² + y² + 2gx + 2fy + c = 0 Read more about Equation of Normal[…]
Let C1, C2 be the centers and r1, r2 be the radii of two circles S = 0 and S’ = 0 respectively. Further represents the line segment from C1 to C2. The following cases arises with regard to the relative position of two circles. 1. C1 + C2 > r1 + r2 In this Read more about Concepts related to two circles[…]
Now we discuss the number of common tangents that exist for the cases specified in the section. Case (i): Each of the given pair of circles lies in the exterior of the other i.e. C₁C₂ > r₁ + r₂. Sub case (i): r₁ ≠ r₂ (r₁, r₂ are radii of the circles)In this case, there Read more about Common Tangents[…]
Pairs of tangents: From a given point two tangents can be drawn to a circle which are real and distinct, coincident or imaginary according as the given point lies outside on or inside the circle.The combined equation of the pair of tangents drawn from point P(x₁, y₁) to the circle x² + y² = a² Read more about Tangents[…]
Intersection of straight line and a circle: Let the equation of circle be x² + y² = a² and the equation of the line be y = mx + c then When points of intersection are real and distinct: Then length of perpendicular from centre should be less the radius. ∴ . When points of Read more about Tangent Equation of Circle & Point of Contact[…]
Square Roots of a Complex Number: Let a + ib be a complex number such that √(a + ib) = x + iy, where x and y are real numbers.If b is positive . If b is negative . Geometrical representation of Conjugate of a Complex Number: Let z = x + iy be a Read more about Square Roots & Geometrical representation of Conjugate of a Complex Number[…]
Addition of Complex Number: Let z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ be two complex numbers. Then their sum z₁ + z₂ is defined as the complex number (a₁ + a₂) + I (b₁ + b₂). Subtraction of Complex Number: Let z₁ = a₁ + ib₁ and z₂ = a₂ + Read more about Algebra & Modulus of complex numbers[…]