If the roots of the quadratic …

According to the question,the given equation is,
(a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0
The discriminant of the given equation is given by
D = [-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}
{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)²
Since 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 {tex}\ne{/tex} 0]
{tex}\Rightarrow{/tex} ad - bc = 0
{tex}\Rightarrow{/tex} ad = bc
{tex} \Rightarrow \frac{a}{b} = \frac{c}{d}{/tex}