Question

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=-7 \cos 2 t, y=-7 \sin 2 t ; 0 \leq t \leq \pi$$

Short answer

Question: Determine the center, radius, and orientation of the circle/arc with the given parametric equations: \(x = -7\cos(2t)\) and \(y = -7\sin(2t)\) for \(0 \leq t \leq \pi\). Also, find the equation in terms of x and y. Answer: The center of the circle is (0, 0), the radius is 7, and the orientation is counterclockwise. The equation in terms of x and y is \(\frac{x^2}{49} + \frac{y^2}{49} = 1\).

Step-by-step solution

Eliminate the parameter

Divide both equations by \(-7\) to obtain the equations in terms of \(\sin(2t)\) and \(\cos(2t)\): $$\frac{x}{-7} = \cos(2t), \frac{y}{-7} = \sin(2t)$$ Now, square both of these expressions: $$\left(\frac{x}{-7}\right)^2 = \cos^2(2t), \left(\frac{y}{-7}\right)^2 = \sin^2(2t)$$ Next, use the trigonometric identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) to eliminate the parameter \(t\): $$\left(\frac{x}{-7}\right)^2 + \left(\frac{y}{-7}\right)^2 = \cos^2(2t) + \sin^2(2t) = 1$$

Find the equation and description in terms of \(x\) and \(y\)

Manipulate the previous equation to find the equation in terms of \(x\) and \(y\): $$\left(\frac{x}{-7}\right)^2 + \left(\frac{y}{-7}\right)^2 = 1$$ Calculating we get: $$\frac{x^2}{49} + \frac{y^2}{49} = 1$$ This is the equation of a circle in terms of \(x\) and \(y\).

Determine the center

In a circle equation \((x-a)^2 + (y-b)^2 = r^2\), the center is \((a, b)\). In this case, since our equation is \(\frac{x^2}{49} + \frac{y^2}{49} = 1\), the equation can also be expressed as \((x-0)^2 + (y-0)^2 = (-7)^2\). Thus, the center of the circle is \((0, 0)\).

Find the radius

Using the equation found in step 2 \((\frac{x^2}{49} + \frac{y^2}{49} = 1\)), we can see that the radius is \(7\).

Indicate the positive orientation

The positive orientation is counterclockwise. In parametric equations, we can determine the orientation by looking at how the angle \(t\) progresses. Since \(t\) goes from \(0\) to \(\pi\) in the given interval, the arc goes counterclockwise. In summary, the circle or circular arc has a center at \((0, 0)\), a radius of \(7\), and follows a counterclockwise orientation. The equation in terms of \(x\) and \(y\) is \(\frac{x^2}{49} + \frac{y^2}{49} = 1\).