Advertisements
Advertisements
प्रश्न
If a, b, c, d are in continued proportion, prove that: (a + d)(b + c) – (a + c)(b + d) = (b – c)2
Advertisements
उत्तर
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. = (a + d)(b + c) – (a + c)(b + d)
= (dk3 + d) (dk2 + dk) – (dk3 + dk) (dk2 + d)
= d(k3 + 1) dk(k + 1) – dk (k2 + 1) d(k2 + 1)
= d2k(k + 1) (k3 + 1) – d2k (k2 + 1) (k2 + 1)
= d2k[k4 + k3 + k + 1 – k4 - 2k2 - 1]
= d2k[k3 – 2k2 + k]
= d2k2[k2 – 2k + 1]
= d2k2(k – 1)2
R.H.S. = (b – c)2
= (dk2 – dk)2
= d2k2(k – 1)2
∴ L.H.S. = R.H.S.
Hence proved.
संबंधित प्रश्न
If `x/a = y/b = z/c` prove that `(2x^3 - 3y^3 + 4z^3)/(2a^3 - 3b^3 + 4c^3) = ((2x - 3y + 4z)/(2a - 3b + 4c))^3`
If (a2 + b2)(x2 + y2) = (ax + by)2; prove that: `a/x = b/y`.
Find two numbers such that the mean proportional between them is 14 and third proportional to them is 112.
Find the fourth proportion to the following:
(p2q - qr2 ), (pqr - pr2 ) and (pq2 - pr2)
Find the two numbers such that their mean proprtional is 24 and the third proportinal is 1,536.
If q is the mean proportional between p and r, prove that
`p^2 - q^2 + r^2 = q^4[(1)/p^2 - (1)/q^2 + (1)/r^2]`.
Find the third proportional to `5(1)/(4) and 7.`
If a, b, c, d, e are in continued proportion, prove that: a : e = a4 : b4.
The America’s famous Golden Gate bridge is 6480 ft long with 756 ft tall towers. A model of this bridge exhibited in a fair is 60 ft long with 7 ft tall towers. Is the model, in proportion to the original bridge?
If two ratios are equal, then they are in ______.
