If the roots of the equation (a^2+b^2)x^2-(ac+bd)x+(c^2+d^2)=0 are equal, prove that ad=bc

If the roots of the equation (a^2+b^2)x^2-(ac+bd)x+(c^2+d^2)=0 are equ

1.	If the roots of the equation (a^2+b^2)x^2-(ac+bd)x+(c^2+d^2)=0 are equal, prove that ad=bc
Answer» According to the question,the given equation is,(a2 + b2)x2 - 2(ac + bd)x + (c2 + d2) = 0The discriminant of the given equation is given byD = [-2(ac + bd)]2 - 4 {tex}\\times{/tex} (a2 + b2) {tex}\\times{/tex} (c2 + d2){tex}\\Rightarrow{/tex} D = 4(ac + bd)2 - 4(a2c2 + a2d2 + b2c2 + b2d2){tex}\\Rightarrow{/tex} D = 4(a2c2 + b2d2 + 2abcd) - 4(a2c2 + a2d2 + b2c2 + b2d2){tex}{/tex}D= 4(2abcd - a2d2 - b2c2){tex}\\Rightarrow{/tex} D = -4[(ad)2 + (bc)2 - 2(ad)(bc)]{tex}\\Rightarrow{/tex} D = -4(ad - bc)2Since the roots of the given equation are given to be equal, therefore,Discriminant, D = 0{tex}\\Rightarrow{/tex} -4(ad - bc)2 = 0{tex}\\Rightarrow{/tex} (ad - bc)2 = 0 [as -4\xa0{tex}\\ne{/tex}\xa00]{tex}\\Rightarrow{/tex} ad - bc = 0{tex}\\Rightarrow{/tex} ad = bcHence Proved.