1. **Problem Statement:** Find the intersection points of the circles given by the equations $$x^2 + y^2 = 1$$ and $$(x-1)^2 + (y-2)^2 = 4$$. 2. **Step 1: Write down the equations clearly.** - Circle 1: $$x^2 + y^2 = 1$$ - Circle 2: $$(x-1)^2 + (y-2)^2 = 4$$ 3. **Step 2: Expand the second circle's equation.** $$(x-1)^2 + (y-2)^2 = 4$$ $$x^2 - 2x + 1 + y^2 - 4y + 4 = 4$$ Simplify: $$x^2 + y^2 - 2x - 4y + 5 = 4$$ $$x^2 + y^2 - 2x - 4y + 1 = 0$$ 4. **Step 3: Subtract the first circle's equation from this to eliminate $x^2 + y^2$.** $$\cancel{x^2 + y^2} - 2x - 4y + 1 = 0 - \cancel{x^2 + y^2} + 1$$ Simplifies to: $$-2x - 4y + 1 = -1$$ 5. **Step 4: Simplify the above equation.** $$-2x - 4y + 1 = -1$$ $$-2x - 4y = -2$$ Divide both sides by -2: $$\cancel{-2}x + \cancel{-4}y = \cancel{-2}$$ $$x + 2y = 1$$ 6. **Step 5: The line $x + 2y = 1$ represents the chord joining the intersection points of the two circles.** 7. **Step 6: Find the intersection points by substituting $x = 1 - 2y$ into the first circle's equation.** $$x^2 + y^2 = 1$$ Substitute: $$(1 - 2y)^2 + y^2 = 1$$ Expand: $$1 - 4y + 4y^2 + y^2 = 1$$ $$5y^2 - 4y + 1 = 1$$ Subtract 1 from both sides: $$5y^2 - 4y = 0$$ 8. **Step 7: Factor the quadratic in $y$.** $$y(5y - 4) = 0$$ So, $$y = 0$$ or $$5y - 4 = 0 \Rightarrow y = \frac{4}{5}$$ 9. **Step 8: Find corresponding $x$ values.** - For $y=0$: $$x = 1 - 2(0) = 1$$ - For $y=\frac{4}{5}$: $$x = 1 - 2\times \frac{4}{5} = 1 - \frac{8}{5} = -\frac{3}{5}$$ 10. **Step 9: Write the intersection points.** $$\boxed{(1, 0) \text{ and } \left(-\frac{3}{5}, \frac{4}{5}\right)}$$ 11. **Step 10: Plausibility check by sketch:** - Circle 1 is centered at $(0,0)$ with radius 1. - Circle 2 is centered at $(1,2)$ with radius 2. - The points $(1,0)$ and $(-\frac{3}{5}, \frac{4}{5})$ lie on both circles. - The line $x + 2y = 1$ passes through these points, confirming it is the chord of intersection. 12. **Step 11: Geometric meaning of the line $x + 2y - 1 = 0$:** - This line is the radical line or chord of intersection of the two circles. - It is the locus of points having equal power with respect to both circles.