Question

A circle with center at \( (x, y) = (0.5, 0) \) and radius = 0.5 intersects with another circle with center at \( (x, y) = (1, 1) \) and radius = 1 at two points. One of the points of intersection \( (x, y) \) is:

• A. \( (0, 0) \)
• B. \( (0.2, 0.4) \)
• C. \( (0.5, 0.5) \)
• D. \( (1, 2) \)

Hint:

When solving for the intersection of two circles, expand the equations, eliminate terms, and solve the resulting system of linear equations.

Answer 1

The Correct Option is B

Given Circles:
The equations of the circles are:
\[ (x - 0.5)^2 + y^2 = 0.5^2 \quad \text{(Equation 1: Circle 1)} \] \[ (x - 1)^2 + (y - 1)^2 = 1^2 \quad \text{(Equation 2: Circle 2)} \]
Step 1: Expanding the Equations
Expanding Equation 1: \[ x^2 - x + y^2 = 0. \] Expanding Equation 2: \[ x^2 - 2x + y^2 - 2y = -1. \]
Step 2: Subtracting the Equations
\[ (x^2 - 2x + y^2 - 2y) - (x^2 - x + y^2) = -1 - 0 \] \[ -x - 2y = -1 \quad \Rightarrow \quad x + 2y = 1 \quad \cdots (3). \]
Step 3: Substituting \( x = 1 - 2y \) in Equation 1
\[ (1 - 2y)^2 - (1 - 2y) + y^2 = 0. \] Expanding: \[ 1 - 4y + 4y^2 - 1 + 2y + y^2 = 0. \] \[ 5y^2 - 2y = 0. \] Factoring: \[ y(5y - 2) = 0. \] Thus, \( y = 0 \) or \( y = 0.4 \).
Step 4: Finding \( x \)
For \( y = 0.4 \): \[ x = 1 - 2(0.4) = 0.2. \]
Final Answer:
One intersection point is \( (0.2, 0.4) \).