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If A, B, C and Dare in Continued Proportion, Then Prove that - ICSE Class 10 - Mathematics

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Question

If a, b, c and dare in continued proportion, then prove that 

(a+ d)(b+ c)-(a+ c)(b+ d)= (b-c)2  

Solution

`"a"/"b" = "b"/"c" = "c"/"d" = "k"`

⇒ c = kd 

b =kc= k2

a= kb= k3

(a+ d)(b + c)-(a+ c)(b + d) = (b- c)2 

LHS

(a+ d)(b + c)-(a+ c)(b + d) 

= ab + bd + ac + cd - ab - bc - ad - cd 

= bd+ ca - bc - ad 

= k2d2 + k4d4 - k3d2 - k3d2

=  k2d2 + k4d4 - 2k3d2

= k2d2 (1+ k2 - 2k) 

RHS

(b - c)2 = (b - c)(b - c) 

= b2 - 2bc + c2

= k4d4 - 2k3d2 + k2d2

= k2d2 (k2 - 2k + 1) 

LHS = RHS. Hence, proved. 

  Is there an error in this question or solution?

APPEARS IN

 Frank Solution for Frank Class 10 Mathematics Part 2 (2016 to Current)
Chapter 9: Ratio and Proportion
Exercise 9.3 | Q: 10.3

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Solution If A, B, C and Dare in Continued Proportion, Then Prove that Concept: Proportions.
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