Advertisements
Advertisements
Question
If a, b, c, d are in continued proportion, prove that: (a + d)(b + c) – (a + c)(b + d) = (b – c)2
Solution
a, b, c, d are in continued proportion
∴ `a/b = b/c = c/d` = k(say)
∴ c = dk, b = ck = dk. k = dk2,
a = bk = dk2. k = dk3
L.H.S. = (a + d)(b + c) – (a + c)(b + d)
= (dk3 + d) (dk2 + dk) – (dk3 + dk) (dk2 + d)
= d(k3 + 1) dk(k + 1) – dk (k2 + 1) d(k2 + 1)
= d2k(k + 1) (k3 + 1) – d2k (k2 + 1) (k2 + 1)
= d2k[k4 + k3 + k + 1 – k4 - 2k2 - 1]
= d2k[k3 – 2k2 + k]
= d2k2[k2 – 2k + 1]
= d2k2(k – 1)2
R.H.S. = (b – c)2
= (dk2 – dk)2
= d2k2(k – 1)2
∴ L.H.S. - R.H.S.
APPEARS IN
RELATED QUESTIONS
If `a/b = c/d` prove that each of the given ratios is equal to
`(5a + 4c)/(5b + 4d)`
Find the mean proportional to (x – y) and (x3 – x2y).
If x and y be unequal and x : y is the duplicate ratio of x + z and y + z, prove that z is mean proportional between x and y.
a, b, c are in continued proportion. If a = 3 and c = 27 then find b.
Find the fourth proportion to the following:
0.7, 4.9 and 1.6
Find the two numbers such that their mean proprtional is 24 and the third proportinal is 1,536.
If a, b, c and d are in proportion, prove that: `(a + c)^3/(b + d)^3 = (a(a - c)^2)/(b(b - d)^2)`
If a, b, c are in continued proportion, prove that: a : c = (a2 + b2) : (b2 + c2)
Write the mean and extreme terms in the following ratios and check whether they are in proportion.
400 gm is to 50 gm and 25 rupees is to 625 rupees
The mean proportional between 4 and 9 is ______.
