Question

If (a+b+c) (a-b+c)=a²+b²+c² show that a,b,c are in continued proportion​

Answer 1

Answer:

QUETION:-

If (a+b+c) (a-b+c)=a²+b²+c² show that a,b,c are in continued proportion.

TO SHOW:-

Show that a,b,c are in continued propotion.

WE HAVE:-

That If A,B,C continued propotion A=a,B=ar and C=$a{r}^{2}$

SO:-

where R is common ratio LHS =

$(a + b + + c)(a - b + c) = {a}^{2} \\ (1 + r + {r}^{2} )(1 - r + {r}^{2} ) = {a}^{2} \\ (1 + r + {r}^{2} - r - {r}^{2} - {r}^{3} + {r}^{2} + {r}^{3} + {r}^{4} </strong><strong>)</strong><strong> \\ = ( {a}^{2} (1 + {r}^{2} + {r}^{4} ) \\ \\ rhs = ( {a}^{2} + {b}^{2} + {c}^{2} ) = {a}^{2} (1 + {r}^{2} + {r}^{4}</strong><strong>)</strong><strong>$

NOW:-

It is proved that LHS=RHS,Hence

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

Thanks :)

Answer 2

ANSWER

To Prove : (a+b+c)(a−b+c)=a

2

+b

2

+c

2

Proof : a,b,c are in continued proportion.

∴

b

a

=

c

b

=k (let)

b=ck

a=bk=(ck)k =ck

2

L.H.S. =(ck

2

+ck+c)(ck

2

−ck+c)

=c

2

(k

2

+k+1)(k

2

−k+1)

=c

2

[(k

2

+1)

2

−(k)

2

]

=c

2

[k

4

+2k

2

+1−k

2

]

=c

2

[k

4

+k

2

+1]

R.H.S. =c

2

k

4

+c

2

k

2

+c

2

=c

2

[k

4

+k

2

+1]

L.H.S = R.H.S.