convert the equation to standard form for: 36x^2+4y^2+144x+48y+144=0

Question

Need to convert the equation to standard form by completing the square on x and y

36x^2+4y^2+144x+48y+144=0

Thanks in advance.

Answer 1

Given equation is 36x^2+4y^2+144x+48y+144 = 0

Add 144 to each side for competing squre.

36x^2+4y^2+144x+48y+144+144 = 0+144

36x^2+144x+144+y^2+48y+144 = 144

36(x^2+4x+4)+4(y^2+12y+36) = 144

36(x+2)^2+4(y+6)^2 = 144

Divide to each side by 144.

36(x+2)^2/144+4(y+6)^2/144 = 144/144

(x+2)^2/4+(y+6)^2/36 = 1

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

Compare it to standerd form of ellipse equation (x-h)^2/b^2+(y-k)^2/a^2 = 1

Where a > b.

 

Answer 2

The equation

To change the expressions (x ²+ 4x ) and (y ²+ 12y ) into a perfect square trinomial,

add (half the x coefficient)² and add (half the y coefficient)² to each side of the equation.

Compare the standard form of ellipse

a ² > b ²

If the larger denominator is under the "y " term, then the ellipse is vertical.center (h, k ).

a  = length of semi-major axis, b  = length of semi-minor axis.

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