Question

what quantity must be added to the term of the ratio p+q:p-q to make it equal to(p+q)^2:(p-q)^2​

Answer 1

$\textbf{To find}$

$\text{The quantity should be added to the terms of}$

$\text{ratio p+q:p-q to make it equal to (p+q)^2:(p-q)^2 }$

$\text{As per given data,}$

$\dfrac{(p+q)+k}{(p-q)+k}=\dfrac{(p+q)^2}{(p-q)^2}$

$\implies[(p+q)+k](p-q)^2=[(p-q)+k](p+q)^2$

$\implies\,(p+q)(p-q)^2+k(p-q)^2=(p-q)(p+q)^2+k(p+q)^2$

$\implies\,k(p-q)^2-k(p+q)^2=(p-q)(p+q)^2-(p+q)(p-q)^2$

$\implies\,k[(p-q)^2-(p+q)^2]=(p-q)(p+q)[(p+q)-(p-q)]$

$\implies\,k[p^2+q^2-2pq-p^2-q^2-2pq]=(p^2-q^2)(2q)$

$\implies\,k(-4pq)=(p^2-q^2)(2q)$

$\implies\,k(-2p)=(p^2-q^2)$

$\implies\,k=\dfrac{p^2-q^2}{-2p}$

$\implies\bf\,k=\dfrac{q^2-p^2}{2p}$

$\therefore\textbf{ \bf\dfrac{q^2-p^2}{2p} should be added}$

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Answer 2

Given:  p+q:p-q

To find: what quantity must be added to the given term to make it equal to (p+q)²:(p-q)²

Solution:

• As we have given that we need to add a term, to make it (p+q)²:(p-q)².

• So let the term be m, we get:

              (a + b) + m/(a - b) + m = (a + b)²/(a - b)²

• Now, here we need to note that m is added to both numerator and denominator as given in the question.

• Now, solving further, we get:

              (a + b)(a - b)²  + (a - b)²m  =  (a + b)²(a - b) + (a + b)²m

• Taking the terms with the coefficient m on one side.

              { (a + b)² - (a - b)² } m  =  (a + b)(a - b)(a - b) - (a - b)(a + b)  (a + b)

              { a² + b² + 2ab - a² - b² + 2ab } m  = (a² - b²)(a - b) - (a² - b²)(a + b)

             (4ab) m =  (a² - b²){a-b-a-b)

             m = (a² - b²){-2b) / 4ab

             m = { b² - a² / 2a }

Answer:

             The quantity that must be added to the term of the ratio p+q:p-q to make it equal to (p+q)²:(p-q)² is { b² - a² / 2a }