Advertisements
Advertisements
प्रश्न
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
उत्तर
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.
संबंधित प्रश्न
If (a – b) : (a + b) = 1 : 11, find the ratio (5a + 4b + 15) : (5a – 4b + 3).
Find duplicate ratio of 3 : 4
A school has 630 students. The ratio of the number of boys to the number of girls is 3 : 2. This ratio changes to 7 : 5 after the admission of 90 new students. Find the number of newly admitted boys.
If a,b,c are in continued proportion,
show that: `("a"^2 + "b"^2)/("b"("a" + "c")` = `("b"("a" + "c"))/("b"^2 + "c"^2)`
Divide Rs 3450 among A, B, and C in the ratio 3 : 5 : 7.
In an army camp, there were provisions for 425 men for 30 days. How long did the provisions last for 375 men?
If `a/b = c/d = e/f`, prove that each ratio is `[(2a^3 + 5c^3 + 7e^3)/(2b^3 + 5d^3 + 7f^3)]^(1/3)`
A line segment 56 cm long is to be divided into two parts in the ratio of 2 : 5. Find the length of each part.
Of the 288 persons working in a company, 112 are men and the remaining are women. Find the ratio of the number of men to the total number of persons.
A typist has to type a manuscript of 40 pages. She has typed 30 pages of the manuscript. What is the ratio of the number of pages typed to the number of pages left?
