Find a polar equation for the curve represented by the given cartesian equation. x + y = 4

Solution:

Given, the cartesian equation is x + y = 4

Converting to polar coordinates,

x = r cosθ

y = r sinθ

Now, x + y = 4 becomes

r cosθ + r sinθ = 4

r (cosθ + sinθ) = 4

Therefore, the polar equation for the curve is r(cosθ + sinθ) = 4.

Example:

Find a polar equation for the curve represented by the given cartesian equation 3x + 4y = 5

Solution:

Given, the cartesian equation is 3x + 4y = 5

Converting to polar coordinates,

x = rcosθ

y = rsinθ

Now, 3x + 4y = 5 becomes

3rcosθ + 4rsinθ = 5

r(3cosθ + 4sinθ) = 5

Therefore, the polar equation for the curve is r(3cosθ + 4sinθ) = 5.

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Find a polar equation for the curve represented by the given cartesian equation. x + y = 4

Summary:

A polar equation for the curve represented by the given cartesian equation x + y = 4 is r(cosθ + sinθ) = 4.