Advertisements
Advertisements
Question
If a, b, c, d are in continued proportion, prove that (b - c)2 + (c - a)2 + (d - b)2 = (d - a)2.
Solution
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
RELATED QUESTIONS
Find the third proportional to a – b and a2 – b2
Find the third proportion to the following :
16x2 and 24x
If x, 12 and 16 are in continued proportion! find x.
If a, b, c and dare in continued proportion, then prove that
`sqrt (("a + b + c")("b + c + d")) = sqrt "ab" + sqrt "bc" + sqrt "cd"`
Show that the following numbers are in continued proportion:
36, 90, 225
35 inland letters cost Rs 87.50. How many such letters can we buy for 315?
If b is the mean proportional between a and c, prove that (ab + bc) is the mean proportional between (a² + b²) and (b² + c²).
If a, b, c and d are in proportion, prove that: (a4 + c4) : (b4 + d4) = a2 c2 : b2 d2.
The length and breadth of a school ground are 150 m and 90 m respectively, while the length and breadth of a mela ground are 210 m and 126 m, respectively. Are these measurements in proportion?
If x + y + z = 0, the value of `x/(y + z)` is ______.
