Two Dimensional Analytical Geometry - II

The Circle: Basic Concepts & Equations

1. Definition of a Circle

A circle is the locus of a point in a plane which moves, such that its distance from a fixed point in the plane is always constant.

• Centre: The fixed point.
• Radius (r): The constant distance.

2. Standard Equation of a Circle

The equation of a circle with centre (h, k) and radius r is:

(x - h)² + (y - k)² = r²

(If the centre is at the origin (0,0), the equation simplifies to x² + y² = r²)

3. General Equation of a Circle

The general form of the circle equation is:

x² + y² + 2gx + 2fy + c = 0

Key Characteristics:

1. It is an equation of degree 2.
2. Coefficient of xy = 0.
3. Coefficient of x² = Coefficient of y² ≠ 0.

Finding Centre and Radius:

• Centre (h, k): C(-g, -f)
• Radius (r): √(g² + f² - c)

Nature of the Circle:

• If g² + f² - c = 0 → Point Circle
• If g² + f² - c > 0 → Real Circle
• If g² + f² - c < 0 → Imaginary Circle

4. Equation with Given Diameter Extremities

If A(x₁, y₁) and B(x₂, y₂) are the end points of a diameter, the equation of the circle is:

(x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0

Two Dimensional Analytical Geometry - II

Further Topics on Circles

5. Family of Circles (Intersection)

The general equation of a circle passing through the points of intersection of a given circle S = 0 and a line L = 0 is:

S + λL = 0

(where λ is a constant parameter)

6. Equation of a Tangent at a Point

To find the equation of a tangent at a point P(x₁, y₁) lying on the circle, apply the following shortcut substitutions to the circle's equation:

• x² → xx₁
• y² → yy₁
• x → (x + x₁) / 2
• y → (y + y₁) / 2

Standard Circle Tangent:

For the circle x² + y² = a², the tangent at (x₁, y₁) is:

xx₁ + yy₁ = a²

General Circle Tangent:

For the circle x² + y² + 2gx + 2fy + c = 0, the tangent at (x₁, y₁) is:

xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

7. Condition for Tangency

For a straight line y = mx + c to be a tangent to the standard circle x² + y² = a², the required condition is:

c² = a²(1 + m²)

8. Point of Contact

If the line y = mx + c touches the circle x² + y² = a², the single point of contact (x₁, y₁) is given by:

P ≡ ( -ma²/c , a²/c )