What conic does the equation #9x^2+16y^2+64y+1=0# represent?

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Answer 1 · 2017-12-08T13:05:26

Please see below.

#9x^2+16y^2+64y+1=0# can be written as

#9x^2+16(y^2+4y+4)-64+1=0#

or #9x^2+16(y+2)^2=63#

or #x^2/7+(y+2)^2/(63/16)=1#

Hence this is an ellipse with center at #(0,-2)#, major axis as #2sqrt7# i.e. #5.29# and minor axis as #2sqrt(63/16)# i.e. #3/2sqrt7# i.e. #3.97#.

Ellipse appears as shown below.

graph{9x^2+16y^2+64y+1=0 [-5.125, 4.875, -4.38, 0.62]}