Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
∵ a, b, c, d are in proportion
`a/b = c/d` = k(say)
a = bk, c = dk.
L.H.S. = `abcd (1/a^2 + 1/b^2 + 1/c^2 + 1/d^2)`
= `bk.b.dk.d [1/(b^2k^2) + 1/b^2 + 1/(d^2k^2) + 1/d^2]`
= `k^2b^2d^2 [(d^2 + d^2k^2 + b^2 + b^2k^2)/(b^2d^2k^2)]`
= d2(1 + k2) + b2(1 + k2)
= (1 + k2)(b2 + d2)
R.H.S. = a2 + b2 + c2 + d2
= b2k2 + b2 + d2k2 + d2
= b2(k2 + 1) + d2(k2 + 1)
= (k2 + 1)(b2 + d2)
∴ L.H.S. = R.H.S.
APPEARS IN
संबंधित प्रश्न
If x + 5 is the mean proportional between x + 2 and x + 9; find the value of x.
Given four quantities a, b, c and d are in proportion. Show that: (a – c)b2 : (b – d)cd = (a2 – b2 – ab) : (c2 – d2 – cd)
Find the value of the unknown in the following proportion :
5 : 12 :: 15 : x
Find the third proportion to the following :
3 and 15
Find the smallest number that must be subtracted from each of the numbers 20, 29, 84 and 129 so that they are in proportion.
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"`
Determine if the following numbers are in proportion:
150, 200, 250, 300
An electric pole casts a shadow of length 20 m at a time when a tree 6 m high casts a shadow of length 8 m. Find the height of the pole.
The ratio of boys and girls in a school is 12 : 5. If the number of girls is 840, the total strength of the school is
If `(a + b)^3/(a - b)^3 = 64/27`
- Find `(a + b)/(a - b)`
- Hence using properties of proportion, find a : b.
