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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 $a?$

 Apr 9, 2021
 #1
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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

 

 

cool cool cool

 Apr 9, 2021

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