Let's convert the rectangular equation [tex]\( y^2 = 4x + 1 \)[/tex] to polar form by following the approaches taken by Anastasia and Maksim.
### Maksim's Approach
Maksim starts with the rectangular equation:
[tex]\[ y^2 = 4x + 1 \][/tex]
He converts this to polar coordinates:
[tex]\[ y = r \sin(\theta) \][/tex]
[tex]\[ x = r \cos(\theta) \][/tex]
Substitute these into the original equation:
[tex]\[ (r \sin(\theta))^2 = 4 (r \cos(\theta)) + 1 \][/tex]
This simplifies to:
[tex]\[ r^2 \sin^2(\theta) = 4r \cos(\theta) + 1 \][/tex]
To isolate [tex]\( r \)[/tex], subtract [tex]\((r \cos(\theta))^2\)[/tex]:
[tex]\[ r^2 - (r \cos(\theta))^2 = 4r \cos(\theta) + 1 \][/tex]
Hence, Maksim's first step:
[tex]\[ r^2 - (r \cos(\theta))^2 \][/tex]
### Anastasia's Approach
Anastasia starts with the same rectangular equation:
[tex]\[ y^2 = 4x + 1 \][/tex]
Similarly, convert this to polar coordinates:
[tex]\[ y = r \sin(\theta) \][/tex]
[tex]\[ x = r \cos(\theta) \][/tex]
Substitute these into the original equation:
[tex]\[ (r \sin(\theta))^2 = 4 (r \cos(\theta)) + 1 \][/tex]
This simplifies to:
[tex]\[ r^2 \sin^2(\theta) = 4r \cos(\theta) + 1 \][/tex]
Hence, Anastasia's first step:
[tex]\[ r^2 \sin^2(\theta) \][/tex]
### Conclusion
Both Maksim and Anastasia have correctly transformed the equation using polar coordinates and their first steps are valid simplifications. Therefore, we can conclude that both are correct:
- Maksim's approach yields the intermediate step: [tex]\[ r^2 - (r \cos(\theta))^2 \][/tex]
- Anastasia's approach yields the intermediate step: [tex]\[ r^2 \sin^2(\theta) \][/tex]
Both approaches handle the conversion accurately:
[tex]\[ \text{both correct} \][/tex]
So, you can state:
You conclude that both are correct because you remember that [tex]\( r^2 \sin^2(\theta) = (r \cos(\theta)) + 4r \cos(\theta) + 1 \)[/tex] is true.
