Unit 2: Circles and Conics - Properties and Tangency

Orthogonal Circles

Two circles are said to be **orthogonal** if they intersect at a right angle (90°). This means that at their point of intersection, the tangents to the two circles are perpendicular.

Let the two circles be:
S₁ ≡ x² + y² + 2g₁x + 2f₁y + c₁ = 0
S₂ ≡ x² + y² + 2g₂x + 2f₂y + c₂ = 0

Their centers are C₁(-g₁, -f₁) and C₂(-g₂, -f₂), and their radii are r₁ = sqrt(g₁² + f₁² - c₁) and r₂ = sqrt(g₂² + f₂² - c₂).

If the circles intersect orthogonally, the triangle formed by the two centers and an intersection point P is a right-angled triangle. By the Pythagorean theorem, (C₁C₂)² = r₁² + r₂².

This leads to the **condition for orthogonality**:

2g₁g₂ + 2f₁f₂ = c₁ + c₂

Radical Axis

The **radical axis** of two circles is the locus of a point from which the lengths of the tangents drawn to the two circles are equal.

Let the two circles be S₁ = 0 and S₂ = 0 (where coefficients of x² and y² are 1).

The equation of the radical axis is:

S₁ - S₂ = 0

Example:
S₁ ≡ x² + y² + 4x + 6 = 0
S₂ ≡ x² + y² - 2x + 1 = 0
Radical Axis: S₁ - S₂ = (x² + y² + 4x + 6) - (x² + y² - 2x + 1) = 0
4x + 6 - (-2x + 1) = 0
6x + 5 = 0 (This is a straight line).

Properties of the Radical Axis:

• It is always a straight line.
• It is perpendicular to the line joining the centers of the two circles.
• If the two circles intersect, the radical axis is their common chord.
• If the two circles touch, the radical axis is their common tangent.

Radical Centre

The **radical centre** of three circles is the single point from which the lengths of tangents drawn to all three circles are equal.

It is found by taking the three circles in pairs and finding their radical axes. The point of intersection of these three radical axes is the radical centre.

Given circles S₁ = 0, S₂ = 0, and S₃ = 0:

1. Find the equation of Radical Axis 1: L₁₂ ≡ S₁ - S₂ = 0
2. Find the equation of Radical Axis 2: L₂₃ ≡ S₂ - S₃ = 0
3. Solve the two linear equations L₁₂ = 0 and L₂₃ = 0 simultaneously.

The resulting point (x, y) is the radical centre. (The third radical axis L₁₃ ≡ S₁ - S₃ = 0 will automatically pass through this point).

Circles Through Intersection

Intersection of Two Circles

Let S₁ = 0 and S₂ = 0 be two intersecting circles.

The equation of *any* circle passing through the intersection points of S₁ and S₂ is given by:

S₁ + kS₂ = 0 (where k is a parameter and k ≠ -1)

If k = -1, the equation becomes S₁ - S₂ = 0, which is the equation of the radical axis (their common chord).

Intersection of a Circle and a Straight Line

Let S = 0 be a circle and L = 0 be a straight line.

The equation of *any* circle passing through the intersection points of S and L is given by:

S + kL = 0 (where k is a parameter)

Condition of Tangency

This is a crucial topic. We want to find the condition for a straight line y = mx + c to just touch (be tangent to) a given conic section.

The method is to substitute y = mx + c into the conic's equation. This will result in a quadratic equation in x. For the line to be a tangent, the two intersection points must be coincident, meaning the quadratic equation must have equal roots.

The condition for equal roots of Ax² + Bx + C = 0 is B² - 4AC = 0.

Applying this method gives the following standard conditions:

Circle: x² + y² = a²

Condition: c² = a²(1 + m²)
Equation of tangent: y = mx ± a * sqrt(1 + m²)

Parabola: y² = 4ax

Condition: c = a/m
Equation of tangent: y = mx + a/m

Ellipse: x²/a² + y²/b² = 1

Condition: c² = a²m² + b²
Equation of tangent: y = mx ± sqrt(a²m² + b²)

Hyperbola: x²/a² - y²/b² = 1

Condition: c² = a²m² - b²
Equation of tangent: y = mx ± sqrt(a²m² - b²)

Unit 2: Exam Quick Tips