8.4 Solve Problems Involving the Intersection of Two Circles

2026 Syllabus Objectives

  • Solve problems involving the intersection of two circles
  • Find points of intersection of two circles
  • Find the equation of a common chord
  • Determine whether two circles intersect, touch, or do not intersect

Understanding Circle Intersections 🔵🔵

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When two circles are positioned in a coordinate plane, they can interact in several ways. Understanding these interactions is crucial for solving problems involving circle geometry.

Types of Circle Interactions

1. Two Distinct Intersection Points

  • The circles cross each other at exactly two points
  • The distance between centres is less than the sum of radii but greater than the difference
  • Condition: ∣r1−r2∣<d<r1+r2|r_1 - r_2| < d < r_1 + r_2, where dd is the distance between centres

2. Circles Touch (Tangent)

  • The circles meet at exactly one point
  • External tangency: circles touch externally
    • Condition: d=r1+r2d = r_1 + r_2
  • Internal tangency: one circle touches the other from inside
    • Condition: d=∣r1−r2∣d = |r_1 - r_2|

3. No Intersection

  • The circles do not meet at any point
  • Separate circles: d>r1+r2d > r_1 + r_2
  • One circle inside the other: d<∣r1−r2∣d < |r_1 - r_2|

Finding Points of Intersection 📌

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To find where two circles intersect, we solve their equations simultaneously.

Method

Step 1: Write both circle equations in standard form

  • Circle 1: (x−a1)2+(y−b1)2=r12(x - a_1)^2 + (y - b_1)^2 = r_1^2
  • Circle 2: (x−a2)2+(y−b2)2=r22(x - a_2)^2 + (y - b_2)^2 = r_2^2

Step 2: Expand both equations if necessary

Step 3: Subtract one equation from the other to eliminate the quadratic terms

Step 4: Solve the resulting linear equation for one variable

Step 5: Substitute back to find the coordinates of intersection points

Key Formula Reminder 🔑

Circle Equation Forms:

  • (x−a)2+(y−b)2=r2(x - a)^2 + (y - b)^2 = r^2 where (a,b)(a, b) is the centre and rr is the radius
  • x2+y2+2gx+2fy+c=0x^2 + y^2 + 2gx + 2fy + c = 0 where (−g,−f)(-g, -f) is the centre and g2+f2−c\sqrt{g^2 + f^2 - c} is the radius

The Common Chord ⚡

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When two circles intersect at two distinct points, the line joining these points is called the common chord.

Finding the Equation of the Common Chord

The equation of the common chord can be found by subtracting one circle equation from the other.

Given:

  • Circle 1: x2+y2+2g1x+2f1y+c1=0x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0
  • Circle 2: x2+y2+2g2x+2f2y+c2=0x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0

Common Chord Equation:

Subtract the two equations:

(x2+y2+2g1x+2f1y+c1)−(x2+y2+2g2x+2f2y+c2)=0(x^2 + y^2 + 2g_1x + 2f_1y + c_1) - (x^2 + y^2 + 2g_2x + 2f_2y + c_2) = 0

This simplifies to:

2(g1−g2)x+2(f1−f2)y+(c1−c2)=02(g_1 - g_2)x + 2(f_1 - f_2)y + (c_1 - c_2) = 0

This is a linear equation representing the common chord.

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