Answer:
The rules for change of coordinates are:
r = √(x^2 + y^2)
θ = tan(y/x)
and:
x = r*cos(θ)
y = r*sin(θ)
1) Now we have the equation:
r = (9/7)*cos(θ) + 2*sin(θ)
Let's multiply both sides by r:
r^2 = r*( (9/7)*cos(θ) + 2*sin(θ) )
r^2 = (9/7)*r*cos(θ) + 2*r*sin(θ)
Now we can replace:
r*cos(θ) by x and r*sin(θ) by y
r^2 = (9/7)*x + 2*y
And we know that:
r = √(x^2 + y^2)
then:
r^2 = x^2 + y^2
So we can replace that in our equation:
x^2 + y^2 = (9/7)*x + 2*y
This is the equation in cartesian coordinates.
2) Now we want to describe the curve.
We can rewrite this as:
[x^2 - (9/7)*x] + [ y^2 - 2*y] = 0
Now we can complete squares:
So we need to add and subtract:
(4.5/7)^2 and 1^2
[x^2 - 2*(4.5/7)*x + (4.5/7)^2 - (4.5/7)^2] + [ y^2 - 2*y + 1 - 1] = 0
(x - (4.5/7) )^2 + (y - 1)^2 - 1 - (4.5/7) = 0
(x - (4.5/7) )^2 + (y - 1)^2 = 1 + 4.5/7
So this is the equation of a circle, centered at:
( 4.5/7, 1) and with a radius √(1 + 4.5/7)
