What is the Equation of a Family of Circles?

S6-SA1-0277

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The equation of a family of circles represents a group of circles that share a common property, like passing through the same two points or touching the same line. Instead of one specific circle, it describes all circles fitting a certain condition using a single equation with a variable parameter.

Simple Example

Quick Example

Imagine you have a specific chai stall in your neighbourhood. If you always buy chai from that stall, that's one specific chai. But if you want to find 'all chai stalls that are within 1 km of your home', that's a family of chai stalls. Similarly, a family of circles describes all circles that fit a certain rule, like all circles that pass through the points (1,1) and (3,3).

Worked Example

Step-by-Step

PROBLEM: Find the equation of the family of circles passing through the intersection of the circle x^2 + y^2 - 4x - 6y + 9 = 0 and the line x + y - 3 = 0.

STEP 1: Identify the given circle (S) and the given line (L).
S: x^2 + y^2 - 4x - 6y + 9 = 0
L: x + y - 3 = 0

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STEP 2: Recall the general form for a family of circles passing through the intersection of a circle S=0 and a line L=0 is S + lambda*L = 0, where lambda is a real number.

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STEP 3: Substitute the equations for S and L into the general form.
(x^2 + y^2 - 4x - 6y + 9) + lambda*(x + y - 3) = 0

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STEP 4: Rearrange the terms to get the standard form of a circle.
x^2 + y^2 + (-4 + lambda)x + (-6 + lambda)y + (9 - 3*lambda) = 0

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ANSWER: The equation of the family of circles is x^2 + y^2 + (-4 + lambda)x + (-6 + lambda)y + (9 - 3*lambda) = 0, where lambda is a parameter.

Why It Matters

Understanding families of circles helps engineers design optimal paths for robots or vehicles, like in self-driving cars. In space technology, it's used to calculate orbits for satellites. Even in medicine, it helps analyze shapes in medical imaging for diagnoses.

Common Mistakes

MISTAKE: Forgetting to use a parameter (like lambda or k) in the equation. | CORRECTION: Always include a parameter (lambda, k, etc.) to represent the 'family' aspect, as it allows for different circles within that family.

MISTAKE: Incorrectly combining the constant terms or coefficients after multiplying by the parameter. | CORRECTION: Be careful with algebraic signs and distribute the parameter (e.g., lambda) correctly to all terms in the line or second circle equation before combining like terms.

MISTAKE: Using the formula S1 + S2 = 0 for a family of circles passing through the intersection of two circles, when it should be S1 + lambda*S2 = 0. | CORRECTION: Remember that S1 + S2 = 0 gives only ONE specific circle (the radical axis), not a family. Always use S1 + lambda*S2 = 0 for a family of circles.

Practice Questions

Try It Yourself

QUESTION: Write the equation of the family of circles passing through the intersection of the circle x^2 + y^2 - 2x = 0 and the line y - 1 = 0. | ANSWER: x^2 + y^2 - 2x + lambda*(y - 1) = 0, or x^2 + y^2 - 2x + lambda*y - lambda = 0

QUESTION: If a family of circles passes through the intersection of x^2 + y^2 - 6x - 8y + 12 = 0 and x^2 + y^2 - 4x - 2y + 4 = 0, write its general equation. | ANSWER: (x^2 + y^2 - 6x - 8y + 12) + lambda*(x^2 + y^2 - 4x - 2y + 4) = 0

QUESTION: Find the equation of the family of circles passing through the origin (0,0) and the intersection of the circles x^2 + y^2 - 2x - 4y = 0 and x^2 + y^2 - 6x - 8y + 10 = 0. | ANSWER: The family equation is (x^2 + y^2 - 2x - 4y) + lambda*(x^2 + y^2 - 6x - 8y + 10) = 0. Since it passes through (0,0), substitute x=0, y=0: (0) + lambda*(10) = 0, which means lambda = 0. So, the specific circle is x^2 + y^2 - 2x - 4y = 0. This means the original equation itself is the answer.

MCQ

Quick Quiz

Which of these represents the general equation of a family of circles passing through the intersection of a circle S=0 and a line L=0?

S = L

S + L = 0

S + lambda*L = 0

lambda*S + L = 0

The Correct Answer Is:

The correct form is S + lambda*L = 0, where lambda is a real number (parameter). This form ensures that for any value of lambda, the equation represents a circle passing through the intersection points.

Real World Connection

In the Real World

Imagine a drone delivering packages in a city. To find the most efficient path around obstacles or to service multiple drop-off points, algorithms might use concepts related to families of circles to calculate possible routes. For example, finding all circular paths that pass through two specific delivery points, like in a Zomato or Swiggy delivery route optimization.

Key Vocabulary

Key Terms

FAMILY OF CURVES: A collection of curves (like circles) that share a common mathematical property and are described by a single equation with a variable parameter. | PARAMETER: A variable (often denoted by lambda, k, or t) in an equation whose value determines the specific curve within a family. | INTERSECTION: The point or points where two or more geometric shapes meet or cross each other. | GENERAL EQUATION: An equation that describes a whole group or family of curves, rather than just one specific curve.

What's Next

What to Learn Next

Great job understanding families of circles! Next, you can explore 'Radical Axis of Two Circles' and 'Coaxial Systems of Circles'. These concepts build directly on what you've learned and will help you solve even more interesting geometry problems!