Question 1180259
Identify the equations of ellipses whose major axis lengths are twice their minor axis lengths.

recall: recall: The equation of an ellipse is
 {{{(x-h)^2/a^2+(y-k)^2/b^2=1}}}-> horizontal major axis, {{{a>b}}}
and
{{{(x-h)^2/b^2+(y-k)^2/a^2=1}}}-> vertical major axis, {{{a>b}}} 

The center is ({{{h}}},{{{k}}}) and the larger of {{{a }}}the major axis ,{{{ b}}} is minor axis.


a. 
{{{4x^2+25y^2+32x-250y+589=0}}}......competing squares you will get

{{{(4x^2+32x)+(25y^2-250y)+589=0}}}

{{{4(x^2+8x)+25(y^2-10y)+589=0}}}

{{{4(x^2+8x+b^2)-4b^2+25(y^2-10y+b^2)-25b^2+589=0}}}

{{{4(x^2+8x+4^2)-4*4^2+25(y^2-10y+5^2)-25*5^2+589=0}}}

{{{4(x+4)^2-64+25(y-5)^2-625+589=0}}}

{{{4(x+4)^2+25(y-5)^2-100=0}}}

{{{4(x+4)^2+25(y-5)^2=100}}}........both sides divide by {{{100}}}

{{{4(x+4)^2/100+25(y-5)^2/100=100/100}}}

{{{(x+4)^2/25+(y-5)^2/4=1}}}

{{{(x+4)^2/5^2+(y-5)^2/2^2=1}}}

{{{h=-4}}},{{{k=5}}}
major axis:{{{a =5}}}
minor axis:{{{b=2}}}

=> major axis length is {{{not}}} twice the  length of minor axis 



b. 

{{{2x^2+8y^2-12x+16y-174=0}}}

{{{(2x^2-12x)+(8y^2+16y)=174}}}

{{{2(x^2-6x+b^2)-2b^2+8(y^2+2y+b^2)-8b^2=174}}}

{{{2(x^2-6x+3^2)-2*3^2+8(y^2+2y+1^2)-8*1^2=174}}}

{{{2(x-3)^2-18+8(y+1)^2-8=174}}}....simplify, both sides divide by {{{2}}}

{{{(x-3)^2-9+4(y+1)^2-4=87}}}

{{{(x-3)^2+4(y+1)^2=87+9+4}}}

{{{(x-3)^2+4(y+1)^2=100}}}........both sides divide by {{{100}}}

{{{(x-3)^2/100+4(y+1)^2/100=100/100}}}

{{{(x-3)^2/100+(y+1)^2/25=1}}}

{{{(x-3)^2/10^2+(y+1)^2/5^2=1}}}

{{{h=3}}},{{{k=-1}}}
major axis:{{{a =10}}}
minor axis:{{{b=5}}}

=> major axis length  {{{is}}} twice the  length of minor axis 


c. 
{{{4x^2+y^2+16x+4y+4=0}}}

{{{(4x^2+16x)+(y^2+4y)=-4}}}

{{{4(x^2+4x+b^2)-4b^2+(y^2+4y+b^2)-b^2=-4}}}

{{{4(x^2+4x+2^2)-4*2^2+(y^2+4y+2^2)-2^2=-4}}}

{{{4(x+2)^2-16+(y+2)^2-4=-4}}}

{{{4(x+2)^2+(y+2)^2=-4+16+4}}}

{{{4(x+2)^2+(y+2)^2=16}}}......both sides divide by {{{16}}}

{{{4(x+2)^2/16+(y+2)^2/16=16/16}}}

{{{(x+2)^2/4+(y+2)^2/16=1}}}

{{{(x+2)^2/2^2+(y+2)^2/4^2=1}}}

{{{h=-2}}},{{{k=-2}}}
major axis:{{{b =4}}}
minor axis:{{{a=2}}}

=> major axis length  {{{is}}} twice the  length of minor axis

d. 

{{{3x^2+12y^2+18x-24y-69=0}}}...simplify, both sides divide by {{{3}}}

{{{x^2+4y^2+6x-8y-23=0}}}

{{{(x^2+6x)+(4y^2-8y)=23}}}

{{{(x^2+6x+b^2)-b^2+4(y^2-2y+b^2)-4b^2=23}}}

{{{(x^2+6x+3^2)-3^2+4(y^2-2y+1^2)-4*1^2=23}}}

{{{(x+3)^2-9+4(y-1)^2-4=23}}}

{{{(x+3)^2+4(y-1)^2=23+9+4}}}

{{{(x+3)^2+4(y-1)^2=36}}}

{{{(x+3)^2/36+4(y-1)^2/36=36/36}}}

{{{(x+3)^2/36+(y-1)^2/9=1}}}

{{{(x+3)^2/6^2+(y-1)^2/3^2=1}}}

{{{h=-3}}},{{{k=1}}}
major axis:{{{a =6}}}
minor axis:{{{b=3}}}

=> major axis length  {{{is}}} twice the  length of minor axis


e. 

{{{16x^2+y^2-64x+8y+16=0}}}

{{{(16x^2-64x)+(y^2+8y)=-16}}}

{{{16(x^2-4x+b^2)-16b^2+(y^2+8y+b^2)-b^2=-16}}}

{{{16(x^2-4x+2^2)-16*2^2+(y^2+8y+4^2)-4^2=-16}}}

{{{16(x-2)^2-64+(y+4)^2-16=-16}}}...simplify, both sides divide by {{{16}}}

{{{(x-2)^2-4+(y+4)^2-1=-1}}}

{{{(x-2)^2+(y+4)^2=-1+4+1}}}
{{{(x-2)^2+(y+4)^2=4}}}.........both sides divide by {{{4}}}

{{{(x-2)^2/4+(y+4)^2/4=1}}}

{{{(x-2)^2/2^2+(y+4)^2/2^2=1}}}


{{{h=2}}},{{{k=-4}}}
major axis:{{{a =2}}}
minor axis:{{{b=2}}}

=> major axis length  {{{is}}} {{{not}}} twice the  length of minor axis


f. 

{{{x^2+9y^2-2x+18y-71=0}}}

{{{(x^2-2x)+(9y^2+18y)=71}}}

{{{(x^2-2x+b^2)-b^2+9(y^2+2y+b^2)-9b^2=71}}}

{{{(x^2-2x+1^2)-1^2+9(y^2+2y+1^2)-9*1^2=71}}}

{{{(x-1)^2-1+9(y+1)^2-9=71}}}

{{{(x-1)^2+9(y+1)^2=71+10}}}

{{{(x-1)^2+9(y+1)^2=81}}}.......both sides divide by {{{81}}}

{{{(x-1)^2/81+9(y+1)^2/81=1}}}

{{{(x-1)^2/81+(y+1)^2/9=1}}}

{{{(x-1)^2/9^2+(y+1)^2/3^2=1}}}

{{{h=1}}},{{{k=-1}}}
major axis:{{{a =9}}}
minor axis:{{{b=3}}}

=> major axis length  {{{is}}} {{{not}}} twice the  length of minor axis