Advertisements
Advertisements
प्रश्न
If a, b, c, d are in continued proportion, prove that (b − c)2 + (c − a)2 + (d − b)2 = (d − a)2.
Advertisements
उत्तर
Since a, b, c, d are in continued proportion, we have
`a/b = b/c = c/d` = K (say)
∴ c = dK, b = cK = dK2 and a = bK = dK3.
L.H.S.
= (b − c)2 + (c − a)2 + (d − b)2
= (dK2 − dK)2 + (dK − dK3)2 + (d − dK2)2
= d2K2(K − 1)2 + d2K2 (1 − K2)2 + d2 (1 − K2)2
= d2 [K2(K − 1)2 + K2(K2 − 1)2 + d2(k2 − 1)2]
= d2 [K2 (K − 1)2 + K2 (K − 1)2 (K + 1)2 + (K − 1)2 (K + 1)2]
= d2 (K − 1)2 [K2 + K2 (K+ 1)2 + (K + 1)2]
= d2 (K − 1)2 [K2 + K2 (K2 + 2K + 1) + K2 + 2K + 1]
= d2 (k − 1)2 [K4 + 2K3 + 3K2 + 2K + 1]
= d2(K − 1)2 (K2 + K + 1)2
= d2[(K − 1) (K2 + K + 1)2]
= d2 (K3 − 1)2 = (dK3 − d)2 = (a − d)2 = (d − a)2 = R.H.S.
Hence, (b − c)2 + (c − a)2 + (d − b)2 = (d − a)2.
APPEARS IN
संबंधित प्रश्न
Find the third proportional to a – b and a2 – b2
If p + r = mq and `1/q + 1/s = m/r`; then prove that p : q = r : s.
Find two numbers whose mean proportional is 18 and the third proportional is 486.
Find the third proportional to:
x - y, x2 - y2
If a : b = c : d, show that (a - c) b2 : (b - d) cd = (a2 - b2 - ab) : (c2 - d2 - cd).
Write (T) for true and (F) for false in case of the following:
32 kg : Rs 36 : : 8 kg : Rs 9
The ratio 92 : 115 in its simplest for is
If Rs 760 is divided between A and B in the ratio 8 : 11, then B's share is
Find the missing number in the box in the proportion:
`8/square = 3.2/4`
If a, b, c and d are in proportion, the value of `(8a^2 - 5b^2)/(8c^2 - 5d^2)` is equal to ______.
