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प्रश्न
If b is the mean proportional between a and c, prove that a, c, a² + b², and b² + c² are proportional.
If b is the mean proportional between a and c, prove that a, c, a² + b², and b² + c² are in proportion.
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उत्तर
∵ b is the mean proportional between a and c, then,
b² = a × c
⇒ b² = ac ...(i)
Now a, c, a2 + b2 and b2 + c2 are in proportion
⇒ `a/c = (a^2 + b^2)/(b^2 + c^2)`
R.H.S.:
`(a^2 + b^2)/(b^2 + c^2)`
Substitute b2 = ac
`(a^2 + b^2)/(b^2 + c^2)`
= `(a^2 + ac)/(ac + c^2)`
= `(a(a + c))/(c(a + c))`
= `a/c`
∴ `a/c = (a^2 + b^2)/(b^2 + c^2)`
Hence proved.
संबंधित प्रश्न
If a, b, c are in continued proportion, show that: `(a^2 + b^2)/(b(a + c)) = (b(a + c))/(b^2 + c^2)`.
If a, b and c are in continued proportion, prove that: a: c = (a2 + b2) : (b2 + c2)
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"`
If a, b, c are in continued proportion, prove that a : c = (a2 + b2) : (b2 + c2).
What number should be subtracted from each of the numbers 23, 30, 57 and 78 so that the remainders are in proportion ?
Find the value of x in each of the following proportions:
x : 92 : : 87 : 116
Show that the following numbers are in continued proportion:
16, 84, 441
If a, b, c, d are in continued proportion, prove that: (a + d)(b + c) – (a + c)(b + d) = (b – c)2
3 : 5 : : `square` : 20
Determine if the following are in proportion.
32, 48, 70, 210
