Home
Class 12
MATHS
If the arithmetic means of two positive ...

If the arithmetic means of two positive number a and b `(a gt b )` is twice their geometric mean, then find the ratio a: b

Text Solution

Verified by Experts

The correct Answer is:
`(2+ sqrt(3)):(2-sqrt(3))`
Let A be the A.M. and G be the G.M. of a and b. Then the numbers a and b are roots of the equation `x^(2)-4Gx+G^(2)=0`b `[because a=2G]`
`rArrx=(4Gpmsqrt(16G^(2)-4G^(2)))/2`
`=(4Gpm2sqrt3G)/2=(2pmsqrt3)G`
Hence, a:b=`(2+sqrt3):(2-sqrt3)`
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERICISE 5.5|10 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERICISE 5.6|11 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERICISE 5.3|9 Videos

  • PROBABILITY II

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|3 Videos

Similar Questions

Explore conceptually related problems

If the arithmetic mean of two positive numbers a & b (a gt b) is twice their geometric mean , then a : b is

Show that the arithmetic means of two positive numbers can never be less than their geometric mean. The arithmetic and geometric means of two positive numbers are 15 and 9 respectively. Find the numbers.

If the A.M. of two positive numbers aa n db(a > b) is twice their geometric mean. Prove that : a : b=(2+sqrt(3)):(2-sqrt(3))dot

The arithmetic mean of two positive numbers is 6 and their geometric mean G and harmonic mean H satisfy the relation G^(2)+3H=48 . Then the product of the two numbers is

Find the arithmetic mean of first n odd positive integers.

Find the geometric mean between 2a and 8a^3

For any two positive numbers a and b,prove that AMgeGM

find the arithmetic mean of the following data.

The arithmetic mean of the numbers 1, 2, 3, ...., n is