Let's examine the rectangular equation and see if it converts correctly in both Dmitry's and Sofie's different approaches to polar coordinates.
The given rectangular equation is:
[tex]\[ x^2 - y^2 = 5y + 3 \][/tex]
First, we convert this equation into polar coordinates. Recall that:
[tex]\[ x = r \cos(\theta) \][/tex]
[tex]\[ y = r \sin(\theta) \][/tex]
Substitute [tex]\( x \)[/tex] and [tex]\( y \)[/tex] into the rectangular equation:
[tex]\[ (r \cos(\theta))^2 - (r \sin(\theta))^2 = 5(r \sin(\theta)) + 3 \][/tex]
Simplify the left-hand side of the equation:
[tex]\[ r^2 \cos^2(\theta) - r^2 \sin^2(\theta) \][/tex]
So, the equation becomes:
[tex]\[ r^2 \cos^2(\theta) - r^2 \sin^2(\theta) = 5r \sin(\theta) + 3 \][/tex]
Now let's compare Dmitry's and Sofie's approaches.
### Dmitry's Approach:
Dmitry’s approach states:
[tex]\[ r^2 = 5 \sin(\theta) + 3 \][/tex]
However, this does not directly correspond to the converted polar form we derived. Therefore, Dmitry's approach does not seem correct.
### Sofie's Approach:
Sofie’s approach states:
[tex]\[ r^2 \cos^2(\theta) - r^2 \sin^2(\theta) = 5(r \sin(\theta)) + 3 \][/tex]
When we compare this with our simplified polar form of the given equation:
[tex]\[ r^2 \cos^2(\theta) - r^2 \sin^2(\theta) = 5r \sin(\theta) + 3 \][/tex]
It is clear that Sofie's approach is an exact match.
Thus, we conclude that Sofie’s approach correctly matches the conversion from the rectangular to the polar form of the given equation.
Therefore, the answer is:
Sofie is correct because her approach matches the conversion from the rectangular to the polar form of the given equation.
