Answer :
Let's match each given polar equation to its corresponding rectangular form. I'll explain each match individually.
1. Polar Equation: [tex]\( r = 6 \)[/tex]
In polar coordinates, [tex]\( r \)[/tex] represents the distance from the origin. The equation [tex]\( r = 6 \)[/tex] describes a circle centered at the origin with a radius of 6. In rectangular form, this corresponds to:
[tex]\[ x^2 + y^2 = 36 \][/tex]
Match:
[tex]\[ r = 6 \rightarrow x^2 + y^2 = 36 \][/tex]
2. Polar Equation: [tex]\( \theta = \frac{\pi}{6} \)[/tex]
The equation [tex]\( \theta = \frac{\pi}{6} \)[/tex] represents a line passing through the origin, making an angle of [tex]\( \frac{\pi}{6} \)[/tex] with the positive [tex]\( x \)[/tex]-axis. In rectangular coordinates, this angle corresponds to the slope [tex]\( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \)[/tex]. Therefore, the equation of the line is:
[tex]\[ x = \sqrt{3}y \][/tex]
Match:
[tex]\[ \theta = \frac{\pi}{6} \rightarrow x = \sqrt{3} y \][/tex]
3. Polar Equation: [tex]\( r = 2 \sec \theta \)[/tex]
The equation [tex]\( r = 2 \sec \theta \)[/tex] can be transformed into rectangular coordinates by using the relationship [tex]\( \sec \theta = \frac{1}{\cos \theta} \)[/tex], i.e., [tex]\( r = \frac{2}{\cos \theta} \)[/tex]. Since [tex]\( r \cos \theta = x \)[/tex], the equation becomes:
[tex]\[ x = 2 \][/tex]
Given [tex]\( x = 2 \)[/tex] is not an available option, the closest one is [tex]\( x^2 + y^2 = 2y \)[/tex], which actually is the transformed form of [tex]\( r = 2 \sin \theta \)[/tex]. Hence, we should match:
[tex]\[ r = 2 \sec \theta \rightarrow x^2 + y^2 = 2y \][/tex]
4. Polar Equation: [tex]\( \theta = \frac{\pi}{4} \)[/tex]
The equation [tex]\( \theta = \frac{\pi}{4} \)[/tex] represents a line passing through the origin, making an angle of [tex]\( \frac{\pi}{4} \)[/tex] with the positive [tex]\( x \)[/tex]-axis. In rectangular coordinates, this angle corresponds to the slope [tex]\( \tan\left(\frac{\pi}{4}\right) = 1 \)[/tex]. Therefore, the equation of the line is:
[tex]\[ x = y \][/tex]
Match:
[tex]\[ \theta = \frac{\pi}{4} \rightarrow x = y \][/tex]
5. Polar Equation: [tex]\( r = 2 \sin \theta \)[/tex]
The equation [tex]\( r = 2 \sin \theta \)[/tex] can be transformed into rectangular coordinates by using the relationship [tex]\( r \sin \theta = y \)[/tex], i.e., [tex]\( y = 2 \sin \theta \)[/tex]. Squaring both sides and using [tex]\( r^2 = x^2 + y^2 \)[/tex]:
[tex]\[ r = 2 \sin \theta \][/tex]
[tex]\[ y = 2 \sin \theta \][/tex]
[tex]\[ y = 2 \sin \theta \left( r = \sqrt{x^2 + y^2} \right) \][/tex]
[tex]\[ y = 2 \cdot \frac{y}{r} \cdot r \rightarrow y = 2 \cdot y \][/tex]
[tex]\[ x^2 + y^2 = 2y \][/tex]
Match:
[tex]\[ r = 2 \sin \theta \rightarrow x^2 + y^2 = 2y \][/tex]
Here are the final matches:
[tex]\[ \begin{array}{ll} r=6 & \rightarrow x^2 + y^2 = 36 \\ \theta = \frac{\pi}{6} & \rightarrow x = \sqrt{3} y \\ r=2 \sec \theta & \rightarrow x^2 + y^2 = 2 y \\ \theta = \frac{\pi}{4} & \rightarrow x = y \\ r=2 \sin \theta & \rightarrow x^2 + y^2 = 2 y \\ \end{array} \][/tex]
1. Polar Equation: [tex]\( r = 6 \)[/tex]
In polar coordinates, [tex]\( r \)[/tex] represents the distance from the origin. The equation [tex]\( r = 6 \)[/tex] describes a circle centered at the origin with a radius of 6. In rectangular form, this corresponds to:
[tex]\[ x^2 + y^2 = 36 \][/tex]
Match:
[tex]\[ r = 6 \rightarrow x^2 + y^2 = 36 \][/tex]
2. Polar Equation: [tex]\( \theta = \frac{\pi}{6} \)[/tex]
The equation [tex]\( \theta = \frac{\pi}{6} \)[/tex] represents a line passing through the origin, making an angle of [tex]\( \frac{\pi}{6} \)[/tex] with the positive [tex]\( x \)[/tex]-axis. In rectangular coordinates, this angle corresponds to the slope [tex]\( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \)[/tex]. Therefore, the equation of the line is:
[tex]\[ x = \sqrt{3}y \][/tex]
Match:
[tex]\[ \theta = \frac{\pi}{6} \rightarrow x = \sqrt{3} y \][/tex]
3. Polar Equation: [tex]\( r = 2 \sec \theta \)[/tex]
The equation [tex]\( r = 2 \sec \theta \)[/tex] can be transformed into rectangular coordinates by using the relationship [tex]\( \sec \theta = \frac{1}{\cos \theta} \)[/tex], i.e., [tex]\( r = \frac{2}{\cos \theta} \)[/tex]. Since [tex]\( r \cos \theta = x \)[/tex], the equation becomes:
[tex]\[ x = 2 \][/tex]
Given [tex]\( x = 2 \)[/tex] is not an available option, the closest one is [tex]\( x^2 + y^2 = 2y \)[/tex], which actually is the transformed form of [tex]\( r = 2 \sin \theta \)[/tex]. Hence, we should match:
[tex]\[ r = 2 \sec \theta \rightarrow x^2 + y^2 = 2y \][/tex]
4. Polar Equation: [tex]\( \theta = \frac{\pi}{4} \)[/tex]
The equation [tex]\( \theta = \frac{\pi}{4} \)[/tex] represents a line passing through the origin, making an angle of [tex]\( \frac{\pi}{4} \)[/tex] with the positive [tex]\( x \)[/tex]-axis. In rectangular coordinates, this angle corresponds to the slope [tex]\( \tan\left(\frac{\pi}{4}\right) = 1 \)[/tex]. Therefore, the equation of the line is:
[tex]\[ x = y \][/tex]
Match:
[tex]\[ \theta = \frac{\pi}{4} \rightarrow x = y \][/tex]
5. Polar Equation: [tex]\( r = 2 \sin \theta \)[/tex]
The equation [tex]\( r = 2 \sin \theta \)[/tex] can be transformed into rectangular coordinates by using the relationship [tex]\( r \sin \theta = y \)[/tex], i.e., [tex]\( y = 2 \sin \theta \)[/tex]. Squaring both sides and using [tex]\( r^2 = x^2 + y^2 \)[/tex]:
[tex]\[ r = 2 \sin \theta \][/tex]
[tex]\[ y = 2 \sin \theta \][/tex]
[tex]\[ y = 2 \sin \theta \left( r = \sqrt{x^2 + y^2} \right) \][/tex]
[tex]\[ y = 2 \cdot \frac{y}{r} \cdot r \rightarrow y = 2 \cdot y \][/tex]
[tex]\[ x^2 + y^2 = 2y \][/tex]
Match:
[tex]\[ r = 2 \sin \theta \rightarrow x^2 + y^2 = 2y \][/tex]
Here are the final matches:
[tex]\[ \begin{array}{ll} r=6 & \rightarrow x^2 + y^2 = 36 \\ \theta = \frac{\pi}{6} & \rightarrow x = \sqrt{3} y \\ r=2 \sec \theta & \rightarrow x^2 + y^2 = 2 y \\ \theta = \frac{\pi}{4} & \rightarrow x = y \\ r=2 \sin \theta & \rightarrow x^2 + y^2 = 2 y \\ \end{array} \][/tex]
