Introduction
A circle is the set of all points in a plane that are at a fixed distance (radius) from a fixed point called the centre. Analytical geometry allows us to represent the circle using coordinate equations, which helps in finding radius, centre, tangents, chords, and intersections.
Standard Form of the Circle
If the centre is (h, k) and radius is r, the equation of the circle is:
(x − h)² + (y − k)² = r²
Circle with Centre at Origin
When centre is (0, 0), the equation becomes:
x² + y² = r²
General Form of the Circle
The general equation of a circle is:
x² + y² + 2gx + 2fy + c = 0
Centre and Radius from General Form
Centre: (−g, −f)
Radius: √(g² + f² − c)
Condition for a General Equation to Represent a Circle
The equation x² + y² + 2gx + 2fy + c = 0 represents a real circle only when:
g² + f² − c > 0
Diameter Form of a Circle
If A(x₁, y₁) and B(x₂, y₂) are endpoints of a diameter, then the equation of the circle is:
(x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0
Parametric Coordinates of a Circle
For a circle with centre (0, 0) and radius r, any point on the circle can be represented as:
(x, y) = (r cos θ, r sin θ)
This form is useful for trigonometric problems and locus-based questions.
Chord of a Circle
A chord is a line segment joining two points on the circle.
Length of Chord
If the distance from centre to chord is d:
Length of chord = 2 √(r² − d²)
Equation of Chord with Midpoint Given
If the midpoint of chord is M(x₁, y₁) and circle has centre (h, k):
(x − x₁)(h − x₁) + (y − y₁)(k − y₁) = 0
Tangent to a Circle
A tangent to a circle is a line that touches the circle at exactly one point.
Tangent at Point (x₁, y₁) on a Circle
For circle x² + y² = r²:
x x₁ + y y₁ = r²
Tangent to General Circle
For circle x² + y² + 2gx + 2fy + c = 0:
At point (x₁, y₁), tangent is:
xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0
Condition for Tangency
For line Ax + By + C = 0 to be tangent to circle x² + y² + 2gx + 2fy + c = 0:
|Ax₀ + By₀ + C| / √(A² + B²) = r
(x₀, y₀) is the centre of the circle.
Normal to a Circle
A normal line at a point on the circle is the line passing through that point and the centre.
Intersection of Line and Circle
To find intersection points:
- Write equation of the line.
- Substitute into circle equation.
- Solve quadratic equation to obtain intersection points.
Position of a Point Relative to a Circle
For point P(x₁, y₁), calculate:
S = x₁² + y₁² + 2gx₁ + 2fy₁ + c
- If S = 0 → point lies on circle
- If S < 0 → point lies inside circle
- If S > 0 → point lies outside circle
Examples (Conceptual Overview)
- Finding centre and radius from general equation
- Writing tangent equation at a given point
- Checking whether a point lies inside or outside a circle
- Finding chord length using perpendicular distance
- Forming circle equation from diameter endpoints
Practice Questions
- Find the centre and radius of the circle: x² + y² − 6x + 4y − 12 = 0.
- Write equation of the circle with centre (3, −2) and radius 5.
- Find the tangent at (4, 3) to the circle x² + y² = 25.
- Determine the position of point (5, 7) relative to circle x² + y² − 10x − 6y + 18 = 0.
Detailed Notes:
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