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
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 `a/b = c/d`, show that: `(a^3c + ac^3)/(b^3d + bd^3) = (a + c)^4/(b + d)^4`.
If a, b and c are in continued proportion, prove that `(a^2 + b^2 + c^2)/(a + b + c)^2 = (a - b + c)/(a + b + c)`
Find the mean proportional between :
17.5 and 0.007
Find the mean proportion of the following :
24 and 6
If ax = by = cz, prove that
`x^2/(yz) + y^2/(zx) + z^2/(xy) = (bc)/a^2 + (ca)/b^2 + (ab)/c^2`.
Find the value of x in the following proportions : 10 : 35 = x : 42
Find the mean proportion of: `(1)/(12) and (1)/(75)`
If 4 : 5 : : x : 35, then the value of x is
If `x/a = y/b = z/c`, prove that `"ax - by"/((a + b)(x- y)) + "by - cz"/((b + c)(y - z)) + "cz - ax"/((c + a)(z - x)` = 3
If a, b, c are in continued proportion, prove that: `(pa^2+ qab+ rb^2)/(pb^2+qbc+rc^2) = a/c`
