IIT JEE , AIEEE Forum - Mathematics , Algebra

srbhvatsa

The roots of the equations (a^2 + b^2 ) - 2 (ac +bd )x + (c^2 +d^2 )=0 are equal if i a:b=c:d ii a:c=b:d iii b:c=a;d iv c:d=a:d and please write the method done to solve this

saarika

the roots of ax2 +bx +c = 0 are equal if the discriminant is zero.

that is b2 -4ac =0

similarly in the given equation

(2(ac+bd))^2 - 4(a^2 + b^2)( c^2 + d^2) =0

on solving we get b^2c^2 +a^2d^2 - 2abcd = 0

=> ( bc-ad)^2 = 0

=> bc = ad => a:c =b:d hence option b is correct

magiclko

any quadratic eequation of the form ax² + bx + c= 0, has roots

x = - b

[b² - 4ac]

2a

whr D =

[b² - 4ac]

now this equaition has

two real unequal roots if D>0

two real equal roots if D=0

and no real roots if D<0

similarly here,

D = 0

=> [2(ac+bd)]² - 4(a² + b²) ( c² + d²) = 0

=> (ac+bd)² - (a² + b²) ( c² + d²) = 0

=> b²c² +a²d² - 2abcd = 0

=> ( bc-ad) = 0

=> a/c = b/d

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