Eliminate m,n from the equations m2x−my+a=0,n2x−ny+a=0,mn+1=0.
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Solution
m2x−my+a=0 ....... (i)
n2x−ny+a=0 ....... (ii)
Multiply both equations, we get (m2x−my+a)(n2x−ny+a)=0 (m2n2x2−m2nxy+am2x−mn2xy+mny2−may+an2x−any+a2)=0
As mn+1=0, we get (x2+mxy+nxy+am2x−y2−may+an2x−any+a2)=0 Since, x(m2−n2)−y(m−n)=0 ....... [Subtracting (ii) from (i)] x(m+n)(m−n)−y(m−n)=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)2x2−ma(m+n)x+an2x−an(m+n)x+a2=0 On simplification, we get x+a=0