A Walk on the Coordinate Plane

The most direct path from $(-1,0)$ to $(1,0)$ that passes through another point $(x,y)$ is a straight line path from $(-1,0)$ to $(x,y)$ followed by a straight line from $(x,y)$ to $(1,0)$ . Thus the curve described in the problem is the locus of all points who's sum of the distances from $(-1,0)$ and $(1,0)$ is $3$ . This curve is an ellipse with foci at $(-1,0)$ and $(1,0)$ , semi-major axis $\frac{3}{2}$ , and semi-minor axis $\sqrt{(\frac{3}{2})^2-1^2}=\frac{\sqrt{5}}{2}$ . So the area enclosed by the ellipse is $\pi(\frac{3}{2})(\frac{\sqrt{5}}{2})=\frac{3\sqrt{5}\pi}{4}\approx 5.269$ .