Question

Convert to polar form y=4x^2

Answer 1

To convert a Cartesian equation like y=4x^2 to polar form, we need to use the substitution:

x = r cos(theta)
y = r sin(theta)

Substituting these expressions into the equation y=4x^2, we get:

r sin(theta) = 4(r cos(theta))^2

Simplifying this equation, we get:

r^2 sin(theta) = 4r^2 cos^2(theta)

Dividing both sides by r^2, we get:

sin(theta) = 4 cos^2(theta)

Using the identity cos^2(theta) + sin^2(theta) = 1, we can solve for cos(theta):

cos(theta) = sqrt(1 - sin^2(theta))

Substituting this expression into the equation sin(theta) = 4 cos^2(theta), we get:

sin(theta) = 4 (1 - sin^2(theta))

Expanding and rearranging this equation, we get:

4sin^2(theta) + sin(theta) - 4 = 0

Using the quadratic formula, we can solve for sin(theta):

sin(theta) = (-1 ± sqrt(17))/8

Since sin(theta) is positive in the first and second quadrants, we take the positive root:

sin(theta) = (-1 + sqrt(17))/8

Substituting this value into the equation cos(theta) = sqrt(1 - sin^2(theta)), we get:

cos(theta) = sqrt(65 - 8sqrt(17))/8

Thus, the polar form of the equation y=4x^2 is:

r sin(theta) = 4r^2 cos^2(theta)

r [(-1 + sqrt(17))/8] = 4r^2 [sqrt(65 - 8sqrt(17))/8]^2

Simplifying and rearranging, we get:

r = 8sqrt(65 - 8sqrt(17))/[(-1 + sqrt(17))]

Answer 2

To convert the equation y = 4x^2 to polar form, substitute the polar coordinates x = rcosθ and y = rsinθ into the equation and simplify.

Step-by-step explanation:

The equation y = 4x^2 can be converted to polar form by substituting the polar coordinates x = rcosθ and y = rsinθ into the equation. Using this substitution, we have:

rsinθ = 4(rcosθ)^2

rsinθ = 4r^2cos^2θ

Dividing both sides by r and simplifying, we get:

sinθ = 4rcos^2θ

Using the identity sin^2θ + cos^2θ = 1, we can rewrite the equation as:

sinθ = 4r(1 - sin^2θ)

Expanding and rearranging, we obtain:

4r(sin^2θ + sinθ) - sinθ = 0

This equation represents the polar form of the given equation y = 4x^2.