Advertisements
Advertisements
प्रश्न
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"`
Advertisements
उत्तर
`"a"/"b" = "b"/"c" = "c"/"d" = "k"`
⇒ c = kd
b =kc= k2d
a= kb= k3d
`sqrt (("a + b + c")("b + c + d")) = sqrt "ab" + sqrt "bc" + sqrt "cd"`
LHS
`sqrt (("a + b + c")("b + c + d"))`
`= sqrt (("k"^3"d" + "k"^2"d" + "kd")("k"^2"d" + "kd" + "d"))`
`= sqrt ("kd" ("k"^2 + "k" + 1) xx "d"("k"^2 + "k" + 1))`
`= sqrt ("kd"^2 ("k"^2 + "k" + 1)^2)`
`= "d" sqrt "k" ("k"^2 + "k" + 1)`
RHS
= `sqrt "ab" + sqrt "bc" + sqrt "cd"`
= `sqrt ("k"^3"d" xx "k"^2"d") + sqrt ("k"^2"d" xx "kd") + sqrt ("kd" xx "d")`
=`sqrt ("k"^5"d"^2) + sqrt ("k"^3"d"^2) + sqrt "k" "d"^2`
= `"k"^2"d" sqrt "k" + "kd" sqrt "k" + "d" sqrt "k"`
= `"d" sqrt "k" ("k"^2 + "k" + 1)`
APPEARS IN
संबंधित प्रश्न
If b is the mean proportion between a and c, show that: `(a^4 + a^2b^2 + b^4)/(b^4 + b^2c^2 + c^4) = a^2/c^2`.
If x and y be unequal and x : y is the duplicate ratio of x + z and y + z, prove that z is mean proportional between x and y.
If a : b : : c : d, then prove that
2a+7b : 2a-7b = 2c+7d : 2c-7d
If p, q and r in continued proportion, then prove the following:
(p2 - q2)(q2 + r2) = (q2 - r2)(p2 + q2)
Find the third proportional to Rs. 3, Rs. 12
If 4 : 5 : : x : 35, then the value of x is
If (a + 2 b + c), (a – c) and (a – 2 b + c) are in continued proportion, prove that b is the mean proportional between a and c.
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?
Write True (T) or False (F) against the following statement:
5.2 : 3.9 : : 3 : 4
If x, y and z are in continued proportion, Prove that:
`x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2) = 1/x^3 + 1/y^3 + 1/z^3`
