Solve the system of equations:
$\begin{aligned}(x-2)^2+y^2 & =1 \\ x^2+4 y^2 & =4\end{aligned}$
We are given a system of two equations with two variables, \(x\) and \(y\):
1) \((x-2)^2 + y^2 = 1\)
2) \(x^2 + 4y^2 = 4\)
The first equation describes a circle centered at \((2, 0)\) with a radius of \(1\). The second equation describes an ellipse centered at \((0, 0)\) with semi-major axis \(2\) along the \(x\)-axis and semi-minor axis \(1\) along the \(y\)-axis. We seek the points \((x, y)\) where these two shapes intersect. Since both equations involve \(x\) and \(y^2\), we can substitute \(y^2\) from one equation into the other to reduce the system to a single-variable quadratic equation.
From the second equation, we have \(4y^2 = 4 - x^2\), which implies \(y^2 = 1 - \frac{x^2}{4}\).
Substituting this into the first equation:
\((x-2)^2 + \left(1 - \frac{x^2}{4}\right) = 1\)
Expanding \((x-2)^2\):
\((x^2 - 4x + 4) + 1 - \frac{x^2}{4} = 1\)
Simplifying the equation:
\(x^2 - 4x + 4 - \frac{x^2}{4} = 0\)
\(\frac{3}{4}x^2 - 4x + 4 = 0\)
Multiplying by \(4\):
\(3x^2 - 16x + 16 = 0\)
We solve the quadratic equation \(3x^2 - 16x + 16 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\(x = \frac{16 \pm \sqrt{(-16)^2 - 4(3)(16)}}{2(3)}\)
\(x = \frac{16 \pm \sqrt{256 - 192}}{6}\)
\(x = \frac{16 \pm \sqrt{64}}{6}\)
\(x = \frac{16 \pm 8}{6}\)
This gives two possible values for \(x\):
\(x_1 = \frac{16 + 8}{6} = \frac{24}{6} = 4\)
\(x_2 = \frac{16 - 8}{6} = \frac{8}{6} = \frac{4}{3}\)
Now we find the corresponding \(y\) values for each \(x\):
If \(x = 4\):
\(4^2 + 4y^2 = 4 \implies 16 + 4y^2 = 4 \implies 4y^2 = -12 \implies y^2 = -3\).
Since \(y\) must be a real number, there are no real solutions for \(y\) when \(x = 4\).
If \(x = \frac{4}{3}\):
\(\left(\frac{4}{3}\right)^2 + 4y^2 = 4 \implies \frac{16}{9} + 4y^2 = 4\)
\(4y^2 = 4 - \frac{16}{9} = \frac{36 - 16}{9} = \frac{20}{9}\)
\(y^2 = \frac{20}{36} = \frac{5}{9}\)
\(y = \pm \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3}\)
Thus, the real solutions are \(\left(\frac{4}{3}, \frac{\sqrt{5}}{3}\right)\) and \(\left(\frac{4}{3}, -\frac{\sqrt{5}}{3}\right)\).
This type of problem is common in geometry and engineering, specifically when analyzing intersections. For instance, if you are designing a mechanical part where a circular gear is moving along a path governed by an elliptical track, determining where they touch helps you calculate friction points or points of contact. Solving such systems allows for precise engineering tolerances and optimal motion control.