1.

If a^2+c^2,ab+cd,b^2+d^2 are in continued propotion prove that a,b,c and d are in propotion

Answer»

Step-by-step explanation:

a,B,c and d are in proportion proved below.

Step-by-step explanation:

GIVEN:

(a^2 + c^2), (ab + cd) and (b^2 + d^2)(a

2

+c

2

),(ab+cd)and(b

2

+d

2

) are in continued proportion.

⇒ (a^2 + c^2) : (ab + cd) = (ab + cd) : (b^2 + d^2)(a

2

+c

2

):(ab+cd)=(ab+cd):(b

2

+d

2

)

⇒ (a^2 + c^2) (b^2 + d^2) = (ab + cd) (ab + cd)(a

2

+c

2

)(b

2

+d

2

)=(ab+cd)(ab+cd)

⇒ a^2b^2 + a^2d^2 + c^2b^2 + c^2d^2 = a^2b^2 + 2abcd + c^2d^2a

2

b

2

+a

2

d

2

+c

2

b

2

+c

2

d

2

=a

2

b

2

+2abcd+c

2

d

2

⇒ a^2d^2 + c^2b^2 -2abcd =0a

2

d

2

+c

2

b

2

−2abcd=0

⇒ (ad -cb)^2 = 0(ad−cb)

2

=0 [a^2d^2 + c^2b^2 -2abcd=(ad -cb)^2[/tex]]

⇒ ad - c = 0

⇒ ad = bc

⇒\frac{a}{b} =\frac{c}{d}

b

a

=

d

c

⇒ a, b, c and d are in proportion.


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