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Question

Eliminate m,n from the equations m2xmy+a=0,n2xny+a=0,mn+1=0.

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Solution

m2xmy+a=0 ....... (i)
n2xny+a=0 ....... (ii)
Multiply both equations, we get
(m2xmy+a)(n2xny+a)=0
(m2n2x2m2nxy+am2xmn2xy+mny2may+an2xany+a2)=0
As mn+1=0, we get
(x2+mxy+nxy+am2xy2may+an2xany+a2)=0
Since, x(m2n2)y(mn)=0 ....... [Subtracting (ii) from (i)]
x(m+n)(mn)y(mn)=0
m+n=yx
Substitute y=(m+n)x in the equation, we get
x2+mx(m+n)x+nx(m+n)x+am2x(m+n)2x2ma(m+n)x+an2xan(m+n)x+a2=0
On simplification, we get
x+a=0

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