To convert the Cartesian equation x² + 3x + 5 + y² - 4y + 2 = 25 into polar form, substitute x = r cos(theta) and y = r sin(theta) into the equation and simplify. The polar form of the given equation is: r² + 3r cos(theta) - 4r sin(theta) = 18
To convert the given Cartesian equation x² + 3x + 5 + y² - 4y + 2 = 25 into polar form, let's substitute x = r cos(theta) and y = r sin(theta), where r and theta are polar coordinates.
Substituting these values into the equation, we get:
(r cos(theta))² + 3(r cos(theta)) + 5 + (r sin(theta))² - 4(r sin(theta)) + 2 = 25
Simplifying the equation further:
r² (cos²(theta) + sin²(theta)) + 3r cos(theta) - 4r sin(theta) + 7 = 25
Since cos²(theta) + sin²(theta) = 1, the equation becomes:
r² + 3r cos(theta) - 4r sin(theta) + 7 = 25
Subtracting 7 from both sides:
r² + 3r cos(theta) - 4r sin(theta) = 18
Therefore, the polar form of the given equation is: r² + 3r cos(theta) - 4r sin(theta) = 18