Home
Class 12
MATHS
If a ,b ,c ,da n dp are distinct real nu...

If `a ,b ,c ,da n dp` are distinct real numbers such that `(a^2+b^2+c^2)p^2-2(a b+b c+c d)p+(b^2+c^2+d^2)lt=0,` then prove that `a ,b ,c , d` are in G.P.

Text Solution

Verified by Experts

`(a^(2)+b^(2)+c^(2))p^(2)-2(ab+bc+cd)p+(b^(2)+c^(2)+d^(2))ge0`
`rArr(ap-b)^(2)++(bp-c)^(2)+(cp-d)^(2)ge0` (1)
Since a,b,c,d and p are real, the inequality (1) is possible only when each of the factors is zero, i.e.,
ap-b=0,bp-c=0,cp-d=0
`rArrp=b/a=c/b=d/c`
`rArr` a,b,c,d are in G.P.
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.37|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.38|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.35|1 Videos

  • PROBABILITY II

    CENGAGE ENGLISH|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

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

Similar Questions

Explore conceptually related problems

If a, b, c, d and p are different real numbers such that (a^2+b^2+c^2)p^2-2(a b+b c+c d)p+(b^2+c^2+d^2)lt=0 , then show that a, b, c and d are in G.P.

If a ,\ b ,\ c ,\ d\ a n d\ p are different real numbers such that: (a^2+b^2+c^2)p^(2)-2(a b+b c+c d)p+(b^2+c^2+d^2)lt=0 , then show that a ,\ b ,\ c and d are in G.P.

If a , b , c ,d and p are distinct real numbers such that (1987, 2M) (a^2+b^2+c^2)p^2-2(a b+b c+c d)P+(b^2+c^2+d^2)geq0,t h e na , b , c , d are in AP (b) are in GP are in HP (d) satisfy a b=c d

If a, b, c are distinct real numbers such that a, b, c are in A.P. and a^2, b^2, c^2 are in H. P , then

If a,b,c,d, x are real and the roots of equation (a^2+b^2+c^2)x^2-2(ab+bc+cd)x+(b^2+c^2+d^2)=0 are real and equal then a,b,c,d are in (A) A.P (B) G.P. (C) H.P. (D) none of these

If a, b, c and d are in G.P. show that (a^2+b^2+c^2)(b^2+c^2+d^2)=(a b+b c+c d)^2 .

If a, b, c and d are in G.P. show that (a^2+b^2+c^2)(b^2+c^2+d^2)=(a b+b c+c d)^2 .

If a , b ,c are three distinct positive real numbers in G.P., then prove that c^2+2a b >3ac

If a ,b ,c ,d are distinct integers in an A.P. such that d=a^2+b^2+c^2, then find the value of a+b+c+d

If a ,b ,c ,d are in G.P., prove that a+b,b+c ,c+d are also in G.P.

Home
Profile