+286 Find the constant k so that the equation
4x^2 + 9y^2 - 8x + 54y + k = 2x^2 + 5y^2 - 12x + 34y
represents an ellipse which has an area of 6 \pi.
+130462 Simplify as
2x^2 + 4x + 4y^2 + 20y = -k complete the square on x and y
2(x^2 + 2x + 1) + 4(y^2 + 5y + 25/4) = -k + 2 + 25
2 ( x + 1)^2 + 4( y + 5/2)^2 = 27 - k
(x + 1)^2 / (1/2) + (y + 5/2)^2 / (1/4) = 27 - k divide both sides by (27 -k)
( x + 1)^2 / [ (1/2)(27 -k)] + (y+ 5/2)^2 / [(1/4) (27 -k) ] = 1
a^2 = (27 - k) / 2 b^2 = (27 - k) / 4
a = sqrt [(27 -k) /2 ] b = sqrt [ (27 -k) /4)]
Area = pi * a * b = 6 pi ......so
pi * a * b = 6 pi
a *b = 6
sqrt [ (27 -k) / 2 * (27 -k) /4 ] = 6
(27 - k) / sqrt 8 = 6
(27 - k) / (2sqrt 2) = 6
27 -k = 6 * 2sqrt 2
27 - k = 12sqrt 2
k = 27 - 12sqrt 2
