x² + (y+5)² = 25 in polar coordinates is r = -10sinθ.
Step - by - Step Explanation
What to find?
The polar equation.
Given:
x² + (y+5)² = 25
To find the polar coordinates, substitute x = rcosθ and y=rsinθ into the above.
(rcosθ)² + (rsinθ + 5)² = 25
Open the parenthesis.
r²cos²θ + r²sin²θ + 10rsinθ + 25 = 25
Subtract 25 from both-side of the equation.
r²cos²θ + r²sin²θ + 10rsinθ + 25 - 25= 25 - 25
r²cos²θ + r²sin²θ + 10rsinθ =0
Factor r²
r²(cos²θ + sin²θ) + 10rsinθ =0
Know that cos²θ + sin²θ=1
r² + 10rsinθ =0
Factor out r
r(r + 10sinθ) = 0
Thus, r+10sinθ = 0
Subtract 10sinθ from both-side
r = -10sinθ
Therefore, x² + (y+5)² = 25 in polar coordinates is r = -10sinθ.