Find the orthogonal trajectories of the family of circles x² + (y - c)² = c², where c is a parameter.
To find the orthogonal trajectories of the given family of circles
\[
x^2 + (y - c)^2 = c^2
\]
where \( c \) is a parameter, we proceed as follows:
### Step 1: Differentiate the Given Family
Rewriting the given equation:
\[
x^2 + y^2 - 2cy + c^2 = __________ _________ ____ _____ ______ ___.
_______ _________ _____ _________ ______ ________ _____ _________ ____ _________ _____ ___.
_________ _______ ______ ____ ________ ________ __________ ______ _____.
______ ______ _____ ___ _____ ________ __________ _________ ____ ___.
_____ _____ ___ __________ _________ _______ _______ ___.
_________ ______ ________ _______ ___ ________ _______ __________ _____ _____ ____ ________.
______ _____ ________ ________ _______ __________ ________ _________ __________ ____.
________ ___ _____ ______ __________ ______ __________ _________ ______ ____.
_________ ______ _____ __________ __________ ___ _____ _______ ____ _________ _____.
______ ____ ____ ____ ____ _______ ____ _______ ____ _____ __________ ___.
____ ______ ____ _________ _________ _______ ______ ___ __________ ________ __________.
__________ _______ _________ ______ ______ ______ ________ ____ ______.
______ ___ ______ ____ _________ _____ ______ _________ ____ ______ _______.
______ _________ ___ ____ _________ __________ ___ ____ ____ _______ _______.
_________ ___ _______ ________ ________ _________ ___ _______ _____ _____ __________ ________.
_________ ______ _________ ______ ____ ______.
__________ ______ ____ ___ _________ _______ ___ ____.
__________ _________ _______ ________ _________ __________ ______ _________ ___ ______ _________ ___.
______ _________ ___ ____ _____ ___ ______ _____ _______ ______ _________ ____.
_____ ______ ____ _________ ________ _____ ____ ___ ____ _____ ________ _______.
___ ___ ____ _________ _________ ___.
_____ ________ _______ _________ ________ ________.
_______ ________ ___ _________ _____ _____ ________ __________ _________ ___ ________ ______.
______ __________ ___ ___ ________ ___ __________ ______.
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