Two circles are drawn, as shown. Each is centered at the origin and the radii are 1 and 2. Determine the value of $k$ so that the shaded region above the $x$-axis, below the line $y=k x$ and between the two circles has an area of 2.

![](https://cdn.mathpix.com/cropped/2024_04_30_d1e45f8438f865d01c92g-2.jpg?height=385&width=442&top_left_y=1577&top_left_x=1362)
Let $\theta$ be the angle (in radians) between the line with equation $y=k x$ and the $x$-axis.

Then $k=\tan \theta$, since the tangent function can be viewed as measuring slope.

The shaded area is equal to the area of a sector with central angle $\theta$ of the larger circle with radius 2 minus the area of a sector with central angle $\theta$ of the smaller circle with radius 1 .

Therefore, $\frac{\theta}{2 \pi} \pi\left(2^{2}\right)-\frac{\theta}{2 \pi} \pi\left(1^{2}\right)=2$.

Multiplying both sides by 2 and dividing out the factor of $\pi$ in numerators and denominators, we obtain $4 \theta-\theta=4$ and so $3 \theta=4$ or $\theta=\frac{4}{3}$.

Thus, $k=\tan \frac{4}{3}$.

ANSWER: $\tan \frac{4}{3}$