Write the rectangular equation (x+7)^+y^=49 in polar form.
You should know that
r^2 = x^2 + y^2 and sinØ = y/r , and cosØ = x/r
so for
(x+7)^+y^=49
x^2 + 14x + 49 + y^2 = 49
r^2 + 14rcosØ = 0
or
r(r + 14cosØ) = 0 or r^2 = - 14r cosØ
or
r = 14cosØ , r ≠ 0
To convert the given equation from rectangular form to polar form, we need to replace the variables x and y with their respective polar coordinate expressions.
In polar coordinates, the relationship between x and y is:
x = r * cos(θ)
y = r * sin(θ)
Substituting these expressions into the given equation:
(x + 7)^2 + y^2 = 49
(a polar coordinate expression for x) + 7)^2 + (a polar coordinate expression for y)^2 = 49
(r * cos(θ) + 7)^2 + (r * sin(θ))^2 = 49
Expanding the equation:
(r^2 * cos^2(θ) + 14r * cos(θ) + 49) + (r^2 * sin^2(θ)) = 49
Simplifying:
r^2 * cos^2(θ) + 14r * cos(θ) + r^2 * sin^2(θ) = 0
r^2 * (cos^2(θ) + sin^2(θ)) + 14r * cos(θ) = 0
r^2 + 14r * cos(θ) = 0
This is the equation in polar form, where r represents the distance from the origin and θ represents the angle.
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