- #1
knowLittle
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Homework Statement
(x+y)^2 + (y-2)^2 =4
2. Homework Equations
y^2 = (2-x)y - (x^2)/2
Equation of an ellipse:
##\left( \dfrac {x-h} {a}\right) ^{2}+\left( \dfrac {y-g} {b}\right) ^{2}=1##
From this, we know that (h,g) is the center of the ellipse.
and the radius along the x and y-axis is a and b.
3. The Attempt at a Solution
I assume that y and x in parametric form is analogous than for a circle, but in this case the radius is not equal everywhere and the center is not the origin.
##y=h+b\sin \theta##
##x=g+a\cos \theta##
dx = -a sin(theta) d(theta)
Area = INT(pi/2 to 0)[-b sin(theta) * a sin(theta) * d(theta)]
= INT(0 to pi/2)[absin^2(theta) * d(theta)]
= INT(0 to pi/2)[(ab/2)(1-cos(2theta) * d(theta)]
= (ab/2)[(theta) - (1/2)sin(2theta)] from 0 to pi/2
= (ab/2)[pi/2 - 0]
= pi*ab/4
The total area of the ellipse will be 4 times this area, so:
Area of ellipse = pi*ab
However, I do not know how to acquire a and b from
the expanded equation of the ellipse:
y^2 = (2-x)y - (x^2)/2
(x^2)/2 +y^2= (2-x)y
Can I say that the radius along the x-axis is sqrt(2) and the radius along the y-axis is 1?
In other words a=sqrt(2) and b=1. Then,
(2-x)y=1
2y-xy=1, but I don't know how to interpret this.
Please help.