Advertisements
Advertisements
Question
If a, b, c, d are in continued proportion, prove that: (a2 – b2) (c2 – d2) = (b2 – c2)2
Advertisements
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. = (a2 – b2) (c2 – d2)
= [(dk3)2 – (dk2)2] [(dk)2 – d2]
= (d2k6 – d2k4) (d2k2 – d2)
= d2k4 (k2 – 1) d2(k2 – 1)
= d4k4 (k2 – 1)2
R.H.S. = (b2 – c2)2
= [(dk2)2 – (dk)2]2
= [d2k2 – d2k2]2
= [d2k2 (k2 – 1)]2
= d4k4(k2 – 1)2
∴ L.H.S. = R.H.S.
APPEARS IN
RELATED QUESTIONS
What quantity must be added to each term of the ratio a + b: a - b to make it equal to (a + b)2 : (a - b)2 ?
Find the third proportional to:
`a/b + b/c, sqrt(a^2 + b^2)`.
If a, b, c, d are in continued proportion, prove that:
`sqrt(ab) - sqrt(bc) + sqrt(cd) = sqrt((a - b + c) (b - c + d)`
Find the value of x in the following proportions : 3 : x = 24 : 2
If a + c = mb and `1/b + 1/d = m/c`, prove that a, b, c and d are in proportion.
If a, b, c and d are in proportion, prove that: `abcd [(1/a^2 + 1/b^2 + 1/c^2 + 1/d^2]` = a2 + b2 + c2 + d2
If 7 : 5 is in proportion to x : 25, then ‘x’ is
Fill the boxes using any set of suitable numbers 6 : `square` : : `square` : 15
Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios form a proportion.
2 kg : 80 kg and 25 g : 625 g
What number must be added to each of the numbers 4, 6, 8, 11 in order to get the four numbers in proportion?
