Advertisements
Advertisements
Question
If b is the mean proportion between a and c, show that:
`("a"^4 + "a"^2"b"^2 + "b"^4)/("b"^4 + "b"^2"c"^2 +"c"^4) = "a"^2/"c"^2`
Advertisements
Solution
Given that b is the mean proportion between a and c.
`"a"/"b" = "b"/"c"` = k
⇒ b = ck ; a = bk = (ck)k = ck2
L.H.S = `("a"^4 + "a"^2"b"^2 + "b"^4)/("b"^4 + "b"^2"c"^2 +"c"^4)`
= `(("ck"^2)^4 + ("ck"^2)^2 ("ck")^2 + ("ck")^4)/(("ck")^4 + (ck)^2"c"^2 + "c"^4)`
= `("c"^4"k"^8 + "c"^4"k"^6 + "c"^4"k"^4)/("c"^4"k"^4 + "c"^4"k"^2 + "c"^4)`
= `("c"^4"k"^4("k"^4 + "k"^2 + 1))/("c"^4("k"^4 + "k"^2 + 1)`
= k4 ..........(i)
R.H.S = `"a"^2/"c"^2 = ("ck"^2)^2/"c"^2 = ("c"^2"k"^4)/"c"^2 = "k"^4`.......(ii)
From (i) and (ii) we get
L.H.S = R.H.S
⇒ `("a"^4 + "a"^2"b"^2 + "b"^4)/("b"^4 + "b"^2"c"^2 +"c"^4) = "a"^2/"c"^2`
Hence proved.
RELATED QUESTIONS
If m : n = 3 : 8, find the value of (3m+2n) : (5m+n)
If (3x-4) : (2x+5) is the duplicate ratio of 3 : 4 , find x.
Divide Rs 1,290 into A, B and C such that A is `2/5` of B and B : C = 4 : 3.
Find the ratio between 3.5 m, 475 cm and 2.8 m
The income of a man is increased in the ratio of 10 : 11. If the increase in his income is Rs 600 per month, find his new income.
In an examination, the number of those who passed and the number of those who failed were in the ratio of 3 : 1. Had 8 more appeared, and 6 less passed, the ratio of passed to failures would have been 2 : 1. Find the number of candidates who appeared.
Find the ratio of 90 cm to 1.5 m.
An alloy contains only zinc and copper and they are in the ratio of 7 : 9. If the weight of the alloy is 8 kg, then find the weight of copper in the alloy.
Present age of father is 42 years and that of his son is 14 years. Find the ratio of age of the father to the age of son, when son was 12 years old.
Given below is a Mathematical term.
Find the ratio of consonants to vowels in the term.
