Find dxdy, if x=acosθ,y=asinθ.
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Step-by-Step Solution
Step 1: Differentiate x with respect to θ
We are given x as a function of θ. To find dθdx, we differentiate acosθ with respect to θ. The derivative of cosθ is −sinθ, so dθdx=−asinθ.
Step 2: Differentiate y with respect to θ
Similarly, we differentiate y with respect to θ. The derivative of asinθ with respect to θ is acosθ, since the derivative of sinθ is cosθ.
Step 3: Apply the Chain Rule
To find dxdy, we use the chain rule for parametric equations. This rule states that dxdy can be found by dividing dθdy by dθdx.
Step 4: Substitute and Simplify
Substitute the expressions for dθdy and dθdx that we found in the previous steps. The a terms cancel out, and sinθcosθ simplifies to cotθ, resulting in −cotθ.