An ellipse passes through the points (0,-1), (2,0), (-2,2) and encloses minimum possible area. Find the equation of the ellipse.
The ellipse sought is the
Steinier Circumellipse, which has the same center as the centroid of the triangle.
The centroid is easily calculated as ([0+2+-2]/3, [-1+0+2]/3) = (0, 1/3). An ellipse centered at that point has a general form of Az^2 + Bx(y-1/3) + C(y-1/3)^2 = 1.
Plugging in each point creates a system of equations:
0*A + 0*B + (16/9)*C = 1
4*A - (2/3)*B + (1/9)*C = 1
4*A - (10/3)*B + (25/9)*C = 1
Solving this system yields A=21/64, B=9/16, C=9/16. Substituting and simplifying to standard form 21x^2 + 36xy + 36y^2 - 12x - 24y - 60 = 0.
Solution