Question

If a,b,c,d are in continuef proportion then show that (b-c)²+(a-c)²+(b-d)²=(a-d)².

Answer 1

Answer :
We know when a , b ,c , d are in continued proportion , we get
ab = bc = cd
Let
ab = bc = cd = k  , Then  a  = bk   , b  = ck   And  c  = dk 
∴ b  =ck = dk.k   = dk​2   , a  = bk  = dk2.k   = dk3

∴ a = dk3  , b  = dk2  ,  c  = dk
We have to prove
( b -c )2  + ( c -a )2  + ( b - d )2   = ( a - d )2

Taking L.H.S. and substitute all values As :

⇒ ( dk2 - dk )2 + (  dk - dk3 )2  + ( dk2 - d )2
⇒[ d2k4  + d2k2  - 2d2k3  ]  + [ d2k2  + d2k6  - 2d2k4  ] + [ d2k4 + d2 - 2d2k2 ]

⇒ d2k4 + d2k2 - 2d2k3   +  d2k2 + d2k6 - 2d2k4   + d2k4 + d2 - 2d2k2

⇒d2 + d2k6 - 2d2k3                ---------------- ( 1 )
Now taking R.H.S. And substitute values , we get
⇒( dk3- d )2

⇒[ d2k6+ d2 - 2d2k3]

⇒d2 + d2k6 - 2d2k3                ---------------- ( 2 )

Now we can see from equation 1 and 2 that

L.H.S.   = R.H.S.                                                   ( Hence proved  )