Question

If the equation (a^2+b^2)-2(ac+bd)x+c^2+d^2=0 has equal roots then a)ab=cd b)ad=bc c)ad=underroot bc d)ab=underroot cd

Answer 1

Answer:

If the equation has equal root

SUM of root = - f/e here e= (a^2+b^2)

f= -2(ac+bd) and g=(c^2+d^2)

Produt of root = g/e

If r and s are the roots

For equal roots r=s

SUM of root = r+s = r+r = 2r

2r= -f/e =2(ac+bd)/ (a^2+b^2)

r= (ac+bd) /(a^2+b^2). (1)

For products of root r*s= r*r = r^2

r^2=g/e= (c^2+d^2)/(a^2+b^2). (2)

Take a square of equation 1

r^2= (ac+bd) ^2/ (a^2+b^2)^2

Now put it in eqution 2

(ac+bd) ^2/ (a^2+b^2)^2=(c^2+d^2)/(a^2+b^2)

(ac+bd) ^2/ (a^2+b^2)=(c^2+d^2)

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

(ac)^2+(bd)^2+2abcd= (ac)^2 +(bc) ^2+(ad)^2+(bd)^2

2abcd= (ad) ^2+(bc)^2

(ad) ^2+(bc)^2- 2abcd= 0

( ad-bc)^2=0

ad-bc=0

ad= bc

So, the (b) option is correct

Answer 2

Answer:

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