The graph of the equation \[ x^2 + 4y^2 - 10x + 58y = k\] is a non-degenerate ellipse if and only if $k > a.$ What is $a?$
+128408 Rearrrange as
x^2 -10x + 4y^2 + 58y = k complete the square on x and y
x^2 -10x + 25 + 4( y^2 + (58/4)y + 841 / 16) = k + 25 + 841 / 4
factor and simplify
(x - 5)^2 + 4 (y + 27/4)^2 = k + 941/4
To be non-degenerate k + 941/4 must be greater than 0
So
k + 941/4 > 0
k > -941/4 = a
