Question

The area bounded by the circles \(\mathrm{x}^{2}+\mathrm{y}^{2}=1, \mathrm{x}^{2}+\mathrm{y}^{2}=4\) and the pair of lines \(\sqrt{3}\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)=4 \mathrm{xy}\) is equal to \(\ldots \ldots \ldots\) (a) \((\pi / 4)\) (b) \((\pi / 2)\) (c) \((5 / 2)\) (d) 3

Short answer

The area bounded by the circles and the pair of lines is equal to \(\frac{1}{2}\pi\).

Step-by-step solution

Analyze Circle Equations

Given two circles with equations: 1. \(x^2 + y^2 = 1\) (Circle 1) 2. \(x^2 + y^2 = 4\) (Circle 2) Circle 1 is centered at the origin (0,0) with radius 1, and Circle 2 is also centered at the origin with radius 2.

Analyze Line Equations

Given the pair of lines equation: \(\sqrt{3}(x^2 + y^2) = 4xy\) Rewrite the equation in terms of y to get: \(y = \frac{\sqrt{3}(x^2 + y^2)}{4x}\) \(\Rightarrow y^2 = \frac{3(x^2 + y^2)}{16x^2}\) \(\Rightarrow y^2 = \frac{3x^2 + 3y^2}{16x^2}\) \(\Rightarrow 16x^2y^2 = 3x^2(13y^2 - 3x^2)\) Since \(x^2 \neq 0\), we can divide both sides by \(x^2\), which simplifies the equation to: \(16y^2 = 3(13y^2 - 3x^2)\) Now, we will find the points of intersection between the circle and the lines.

Find Points of Intersection

To find the points of intersection between the circles and lines, we will substitute one equation into the other. From the equation of Circle 1, we have: \(y = \sqrt{1 - x^2}\) Plug this into the line equation to get: \(16(1 - x^2) = 3(13 - 3x^2)\) Simplifying, we get: \(16 = 39x^2\) \(\Rightarrow x^2 = \frac{16}{39}\) Now, let's find the corresponding y-coordinates by plugging back into the Circle 1 equation: \(y^2 = 1 - x^2 = 1 - \frac{16}{39} = \frac{23}{39}\) \(\Rightarrow y = \pm \sqrt{\frac{23}{39}}\) Hence, the points of intersection are \(\left(\pm \sqrt{\frac{16}{39}}, \pm \sqrt{\frac{23}{39}}\right)\).

Set up the Integration

To find the area enclosed by the circles and the lines, we will integrate the difference between their radial functions over the angles for each region. For Circle 1, the angle swept out by the lines is given by: \(\alpha = 2 \arctan \frac{\sqrt{23}}{4}\), then the area enclosed by Circle 1 and lines is given by: \(A_1 = \int_{-\alpha}^{\alpha} \frac{1}{2} r_1^2 d\theta = \int_{-\alpha}^{\alpha} \frac{1}{2} d\theta\) For Circle 2, the angle swept out by the lines is given by: \(\beta = 2 \arctan \frac{\sqrt{23}}{16}\), then the area enclosed by Circle 2 and lines is given by: \(A_2 = \int_{-\alpha}^{\alpha} \frac{1}{2} r_2^2 d\theta = \int_{-\alpha}^{\alpha} 2 d\theta\) Thus, the total area enclosed (\(A_{total}\)) is given by: \(A_{total} = A_2 - A_1\)

Evaluate the Integral and Choose the Correct Answer

Evaluating the integrals, we get: \(A_{total} = \left(2 \times 2\arctan\frac{\sqrt{23}}{16} - \frac{1}{2} \times 2\arctan\frac{\sqrt{23}}{4}\right) = \frac{3}{2} \times 2\arctan\frac{\sqrt{23}}{4}\) Since the options are given in terms of \(\pi\), we can find a relationship between the angle \(\alpha\) and \(\pi\). We know that \(\tan\alpha = \frac{\sqrt{23}}{4}\), so \(\sin\alpha = \frac{\sqrt{23}}{\sqrt{401}}\) and \(\cos\alpha = \frac{4}{\sqrt{401}}\). So, the half-angle formula for \(\tan\) gives us: \(\tan\frac{\alpha}{2} = \frac{\sin\alpha}{1 + \cos\alpha}\) Plugging in the values, we get: \(\tan\frac{\alpha}{2} = \frac{\sqrt{3}}{3}\) Now we can rewrite our integral in terms of \(\pi\), and evaluate the total area: \(A_{total} = \frac{3}{2}(2\arctan\frac{\sqrt{23}}{4}) = \frac{3}{2}(2\frac{\pi}{6})\) \(A_{total} = \frac{3}{2}(\frac{\pi}{3}) = \frac{1}{2}\pi\) So our answer is (b) \(\frac{1}{2}\pi\).

Key concepts

Circles

Understanding circles is crucial in coordinate geometry, especially when dealing with areas bound by various geometric shapes. Circles are defined by their equations in the form \(x^2 + y^2 = r^2\), where \((x, y)\) represents any point on the circle, and \(r\) is the radius.
In this exercise, we have two circles:

• Circle 1: \(x^2 + y^2 = 1\), with a radius of 1.
• Circle 2: \(x^2 + y^2 = 4\), with a radius of 2.

Both circles are centered at the origin \((0,0)\). This means they are concentric circles because they share the same center. The distance from the center to any point on Circle 1 is always 1, while for Circle 2, it's always 2. Recognizing these properties helps in visualizing how circles interact with other shapes, like lines, to form enclosed regions.

Pair of Lines

When it comes to the pair of lines in this problem, their equation is given in an interesting form: \(\sqrt{3}(x^2 + y^2) = 4xy\). This equation might not look like a typical line equation initially. However, by rearranging and unfolding it, we start seeing its potential intersections with circles.
To derive meaningful information from the given equation, we simplify and rearrange terms to align with more recognizable linear equations. Solving this involves diving into properties unique to coordinate equations where both \(x\) and \(y\) are variables that define the line path. By going through these steps, one can determine where exactly these lines touch or cross the circles.
This equation transforms and results in lines that can intersect the circles at notable points. Understanding the intersection allows us to understand more about the resulting geometric area.

Area of Region

The area of the region bound by multiple geometric figures can be a challenging aspect of coordinate geometry. Here, the goal is finding the area enclosed by two circles and a pair of lines.

• We first recognize the radial boundaries set by the circle equations.
• Next, the angle measures formed by the intersections, represented in terms of \(\theta\), offer integral bounds for computing the area.
• Specific formulas involving integration of radial spans (like \(A_1\) and \(A_2\)) help quantify the areas between those geometric figures.

By evaluating the difference between these integrated values, the desired bound area emerges. Understanding this component is crucial for dissecting how each circle and line contributes to the total enclosed space.

Intersection Points

Critical to understanding the problem is calculating where circles and lines intersect. Such points allow us to map out how spaces between them are formed and how areas add together.
In this exercise, we solved for intersection points by plugging circle equations into the line equation, which yielded specific \(x\) and corresponding \(y\) values:

• \(x = \pm \sqrt{\frac{16}{39}}\)
• \(y = \pm \sqrt{\frac{23}{39}}\)

These coordinates mark where lines cut through the circles, forming distinct intersections. By knowing these points, we can also determine the angles involved, which play a part in calculating the integral or establishing the area's extent.
These spatial cues bridge all major variables (the \(x^2\) and \(y^2\) terms) together and help depict a clearer geometric picture.