convert the polar equation r^2=2cos^2theta + 3sin^2theta into a rectangular equation
Multiply both sides by r^2:
r^4=r^2(2cos^2(theta)+3sin^2(theta))
(r^2)^2=2(r*cos(theta))^2+3(r*sin(theta)^2
(x^2+y^2)^2=2x^2+3y^2
To convert the given polar equation into a rectangular equation, we need to replace the polar coordinates (r, θ) with Cartesian coordinates (x, y).
Let's start by simplifying the given equation. In this case, we notice that we can use the identity sin^2θ + cos^2θ = 1.
r^2 = 2cos^2θ + 3sin^2θ
r^2 = 2(1 - sin^2θ) + 3sin^2θ
r^2 = 2 - 2sin^2θ + 3sin^2θ
r^2 = 2 + sin^2θ
Now, we substitute x = rcosθ and y = rsinθ into the equation.
x^2 + y^2 = 2 + sin^2θ
Since x = rcosθ and y = rsinθ, we can replace r^2 with x^2 + y^2.
x^2 + y^2 = 2 + sin^2θ
Thus, the rectangular equation equivalent to the given polar equation r^2 = 2cos^2θ + 3sin^2θ is x^2 + y^2 = 2 + sin^2θ.