Find the values of a and b in the equations 9/a^2 + 4/b^2 = 1 and 3a√a^2-b^2 = 2b^2.

Question

Use elimination process.

Answer 1

there is some ambiguity as to the scope of the square root sign

assuming it applies to a^2, then sqr(a^2) =a, and

to simplify replace a^2 by x and b^2 by y

let x=a^2, y = b^2

9/a^2 + 4/b^2 = 1

9/x + 4/y = 1

3a^2 -b^2 = 2b^2

3x - y = 2y

3x = 3y

y = x

9/x +4/x =1

9 +4 = x

x=13=y

x=a^2

a^2 = 13

a= +/- sqr13 = b

a=b= sqr13 or -sqr13

9/13 + 4/13

13/13 = 1

another interpretation of the problem is to have the square root sign include a^2+b^2

then it gets complicated:

3sqr(x+y) = 2y

9x+9y = 4y^2

9/x+4/y = 1

9/x= 1-4/y = (y-4)/y

x = 9y/(y-4)

9(9y/(y-4) + 9y = 4y^2

81y + 9y^2-36y = 4y^3 -16y^2

4y^3 -25y^2 -45y = 0

y(4y^2 -25y -45) = 0

y = 0 makes the 1st equation undefined

or y = 25/8+/-(1/8)sqr(625+4(4)(45))

y = (25+/-sqr1345)/8

b=sqr(25+/-sqr1345)/8)

x= 9y/(y-4)

a= sqr(9y/(y-4) substitute (25+/-sqr1345)/8 for y

there's also a possibility the problem was miscopied as a[sqr(a^2), a(sqra)^2, or a[sqr(a^2+b)] are a bit peculiar but the only way to interpret the problem as written