Circles: Equations, Radius and Center

• A circle is the set of all points equidistant from a fixed point called the center.
• The distance from the center to any point on the circle is called the radius.

Equation of a Circle

• The standard equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².
• If the center is at the origin (0, 0), the equation simplifies to x² + y² = r².
• The general equation of a circle is x² + y² + 2gx + 2fy + c = 0, where the center is (-g, -f) and the radius is √(g² + f² - c).
• Completing the square can transform the general equation into the standard equation to find the center and radius.

Finding the Center and Radius

• Given the equation (x - h)² + (y - k)² = r², the center is (h, k) and the radius is r.
• From the general equation x² + y² + 2gx + 2fy + c = 0, the center is (-g, -f), and the radius r = √(g² + f² - c).
• For example, given x² + y² - 4x + 6y - 3 = 0, rewrite as (x - 2)² + (y + 3)² = 16, hence the center is (2, -3) and the radius is 4.

Intersection of a Line and a Circle

• To find where a line intersects a circle, solve the equations simultaneously.
• Substitute the equation of the line into the equation of the circle.
• This results in a quadratic equation; the solutions of this equation determine the points of intersection.
• If the quadratic has two distinct real roots, the line intersects the circle at two points.
• If the quadratic has one repeated real root, the line is tangent to the circle.
• If the quadratic has no real roots, the line does not intersect the circle.

Tangents to a Circle

• A tangent to a circle is a line that touches the circle at only one point.
• The tangent at any point on a circle is perpendicular to the radius at that point.
• To find the equation of a tangent at a given point (x₁, y₁) on the circle (x - h)² + (y - k)² = r², first find the gradient of the radius to the point (x₁, y₁), which is (y₁ - k) / (x₁ - h).
• The gradient of the tangent is the negative reciprocal of the radius's gradient, i.e., m_tangent = -(x₁ - h) / (y₁ - k).
• Use the point-slope form of a line y - y₁ = m_tangent * (x - x₁) to find the equation of the tangent.
• If the circle's center is at the origin, the gradient of the radius to (x₁, y₁) is y₁/x₁, and the gradient of the tangent is -x₁/y₁, giving the tangent equation y - y₁ = (-x₁/y₁) * (x - x₁).
• When finding tangents from an external point, let the tangent point be (x₁, y₁) and use the circle equation and the gradient method to form two equations. Solve these simultaneously.

Normal to a Circle

• The normal to a circle at a point is a line perpendicular to the tangent at that point.
• The normal passes through the center of the circle.
• The equation of the normal can be found using the coordinates of the point on the circle and the center of the circle.
• The gradient of the normal is the same as the gradient of the radius to that point.

Circle Theorems relevant to coordinate geometry

• The perpendicular from the center of a circle to a chord bisects the chord.
• The angle in a semicircle is a right angle.
• The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.

Parametric Equations of a Circle

• For a circle with center (0,0) and radius r, the parametric equations are x = r cos(θ), y = r sin(θ).
• For a circle with center (h,k) and radius r, the parametric equations are x = h + r cos(θ), y = k + r sin(θ).
• Parametric equations can be useful in problems involving angles or motion around a circle.

Length of a Chord

• To find the length of a chord, determine the coordinates of the points where a line intersects the circle by solving their equations simultaneously.
• Use the distance formula to calculate the length of the chord between these two points.

Area of a Sector

• The area of a sector with angle θ (in radians) in a circle of radius r is (1/2)r²θ.
• If the angle is given in degrees, convert it to radians by multiplying by π/180 before using the formula.

Area of a Segment

• The area of a segment is the area of the sector minus the area of the triangle formed by the radii and the chord.
• Area of segment = (1/2)r²θ - (1/2)r²sin(θ), where θ is the angle in radians.

Common Problem-Solving Techniques

• Completing the square to find the center and radius of a circle from its general equation.
• Using the distance formula to calculate distances between points related to the circle.
• Applying simultaneous equations to find intersection points.
• Using trigonometric identities when dealing with parametric equations.
• Utilizing the properties of tangents and normals in geometric problems.