Question

if a b c are in continued proportion show that a2+b2/b(a+c)=b(a+c)/b2+c2 ​

Answer 1

Step-by-step explanation:

Given

a, b, c are in continued proportion.Therefore

$\frac{a}{b} = \frac{b}{c} = k$

$or \: b = ck \: and \: a = bk = c {k}^{2}$

To Prove

$\frac{ {a}^{2} + {b}^{2} }{b(a + c)} = \frac{b(a + c)}{ {b}^{2} + {c}^{2} }$

Proof

LHS =

$\frac{ {a}^{2} + {b}^{2} }{b(a + c)} = \frac{ {c}^{2} {k}^{4} + {c}^{2} {k}^{2}}{ck(c {k}^{2} + c)}$

$= \frac{ {c}^{2} {k}^{2}( {k}^{2} + 1) }{ {c}^{2} k( {k}^{2} + 1 )} = k$

RHS =

$\frac{b(a + c)}{ {b}^{2} + {c}^{2} } = \frac{ck(c {k}^{2} + c)}{ {c}^{2} {k}^{2} + {c}^{2} }$

$= \frac{ {c}^{2}k( {k}^{2} + 1) }{ {c}^{2} ( {k}^{2} + 1)} = k$

LHS = RHS = k

Hence Proved.

Answer 2

Answer:

1=1

Step-by-step explanation:

1. We know if the ratio is continued proportion then b2=ac.
2. Interchange any one variable in two format for example: b2=ac. Write instead of b2, ac.
3. then take common from both side
4. Then do cross multipication.
5. Then divide ac(a+c)2 in both sides.
6. Then write (Showed).