how would you determine a conic represented by the polar equation r=a*cos(x)+b*sin(x). then how would you convert the equation to rectangular form? please help
permit me to change the x in cos(x) to
theta
for a point (r,theta) on the polar graph and that same point as (x,y) on the rectangular coordinate system graph
cos(theta) =x/r and sin(theta) = y/r
and x^2 + y^2 = r^2
so r = acos(theta) + bsin(theta) gives
r = ax/r + by/r
r^2 = ax + by
but how would you convert that into rectangular form?
then r^2 = x^2 + y^2
so
x^2 + y^2 = ax + by
isn't that the form for the equation of a circle?
can you complete the square to find its radius and centre?+
To determine the conic represented by the polar equation r = a*cos(x) + b*sin(x), you can convert it to rectangular form using the following steps:
Step 1: Use the trigonometric identity cos(x) = (r*cos(θ))/r and sin(x) = (r*sin(θ))/r.
Substitute these values in the polar equation:
r = a * ((r * cos(θ)) / r) + b * ((r * sin(θ)) / r)
Step 2: Simplify the equation:
r = a * cos(θ) + b * sin(θ)
Step 3: Replace r with √(x² + y²) and rewrite cos(θ) and sin(θ) using their definitions:
√(x² + y²) = a * (x/√(x² + y²)) + b * (y/√(x² + y²))
Step 4: Square both sides of the equation to eliminate the square root:
x² + y² = a²*x²/(x² + y²) + b²*y²/(x² + y²)
Step 5: Multiply both sides of the equation by (x² + y²) to remove the denominators:
x²(x² + y²) + y²(x² + y²) = a²*x² + b²*y²
Step 6: Expand and simplify the equation:
x⁴ + 2x²y² + y⁴ + x²y² + y²x² = a²x² + b²y²
Step 7: Rearrange the terms to get the equation in the standard form:
x⁴ + y⁴ + (2 + a² + b²) * x²y² - a²x² - b²y² = 0
The resulting equation is in rectangular form and represents the conic.