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we find Discriminant of this equation,
=(2mc)2-4(1+m2)(c2-a2)=0
=4m2c2-(4+4m2)(c2-a2)
=4m2c2-4c2+4a2-4m2c2+4m2a2=0
=4m2a2+4a2-4c2=0
=a2m2+a2=c2
=a2(1+m2)=c2
Hence proved