Use the distributive property :
2r - rcos(theta) = 22r−rcos(θ)=2
2r = rcos(theta) + 22r=rcos(θ)+2
Substitute x for rcos(theta)rcos(θ) and sqrt(x^2 + y^2)√x2+y2 for r:
2sqrt(x^2 + y^2) = x + 22√x2+y2=x+2
Square both sides:
4(x^2 + y^2) = x^2 + 4x + 44(x2+y2)=x2+4x+4
4x^2 + 4y^2 = x^2 + 4x + 44x2+4y2=x2+4x+4
3x^2 - 4x + 4y^2 = 43x2−4x+4y2=4
Add 3h^23h2 to both sides:
3x^2 - 4x + 3h^2 + 4y^2 = 3h^2+ 43x2−4x+3h2+4y2=3h2+4
Remove a factor of the 3 from the first 3 terms:
3(x^2 - 4/3x + h^2) + 4y^2 = 3h^2+ 43(x2−43x+h2)+4y2=3h2+4
Use the middle in the right side of the pattern (x - h)^2 = x^2 - 2hx + h^2(x−h)2=x2−2hx+h2 and middle term of the equation to find the value of h:
-2hx = -4/3x−2hx=−43x
h = 2/3h=23
Substitute the left side of the pattern into the equation:
3(x - h)^2 + 4y^2 = 3h^2+ 43(x−h)2+4y2=3h2+4
Substitute 2/323 for h and insert a -0 into the y term:
3(x- 2/3)^2 + 4(y-0)^2 = 3(2/3)^2+ 43(x−23)2+4(y−0)2=3(23)2+4
3(x- 2/3)^2 + 4(y-0)^2 = 16/33(x−23)2+4(y−0)2=163
Divide both sides by 16/3163
3(x- 2/3)^2/(16/3) + 4(y-0)^2/(16/3) = 13(x−23)2163+4(y−0)2163=1
(x- 2/3)^2/(16/9) + (y-0)^2/(16/12) = 1(x−23)2169+(y−0)21612=1
Write the denominators as squares:
(x- 2/3)^2/(4/3)^2 + (y-0)^2/(4/sqrt(12))^2 = 1(x−23)2(43)2+(y−0)2(4√12)2=1
The is standard Cartesian form of the equation of an ellipse with a center at (2/3,0)(23,0); its semi-major axis is 4/343 units long and is parallel to the x axis and its semi-minor axis is 4/sqrt(12)4√12 units long and is parallel to the y axis.
