How would I rewrite this in polar form? // x^2 + y^2 - 6y - 8 = 0
r^2 = 6r sin theta +8
To rewrite the equation x^2 + y^2 - 6y - 8 = 0 in polar form, we need to substitute x and y with their respective polar coordinate equivalents.
Step 1: Convert x and y in terms of polar coordinates.
Using the conversion formulas:
x = r * cos(theta)
y = r * sin(theta)
Step 2: Substitute x and y in the equation.
(r * cos(theta))^2 + (r * sin(theta))^2 - 6(r * sin(theta)) - 8 = 0
Step 3: Simplify the equation.
r^2 * cos^2(theta) + r^2 * sin^2(theta) - 6r * sin(theta) - 8 = 0
Step 4: Simplify further using trigonometric identities.
r^2(cos^2(theta) + sin^2(theta)) - 6r * sin(theta) - 8 = 0
Since cos^2(theta) + sin^2(theta) equals 1, the equation simplifies to:
r^2 - 6r * sin(theta) - 8 = 0
Therefore, the polar form of the equation x^2 + y^2 - 6y - 8 = 0 is r^2 - 6r * sin(theta) - 8 = 0.
To rewrite the equation x^2 + y^2 - 6y - 8 = 0 in polar form, we can use the conversion formulas between Cartesian coordinates (x, y) and polar coordinates (r, θ).
Step 1: Convert y to r * sin(θ)
Since y appears as y^2 and -6y in the equation, we need to express y in terms of r and θ using the equation r * sin(θ).
y = r * sin(θ)
Step 2: Replace x^2 + y^2 in the equation with r^2
In polar coordinates, x^2 + y^2 is represented as r^2. So substitute x^2 + y^2 in the equation with r^2.
r^2 - 6y - 8 = 0
Step 3: Substitute y with r * sin(θ)
Replace y with r * sin(θ) in the equation.
r^2 - 6(r * sin(θ)) - 8 = 0
Step 4: Simplify the equation
Expand the equation and rearrange it if necessary.
r^2 - 6r * sin(θ) - 8 = 0
This is the rewritten equation of x^2 + y^2 - 6y - 8 = 0 in polar form.
x^2+y^2-6y-8 = 0
r^2 - 6rsinθ - 8 = 0