Home
Class 12
MATHS
If a, b, d and p are distinct non - zero...

If a, b, d and p are distinct non - zero real numbers such that `(a^2+b^2 + c^2) p^2 - 2(ab+bc+cd)p+(b^2 + c^2 +d^2) le 0` then n. Prove that a, b, c, d are in G. P and ad = bc

A

are in AP

B

are in GP

C

are in HP

D

satisfy `ab = cd`

Text Solution

Verified by Experts

The correct Answer is:
B
Here, `(a^(2) + b^(2) + c^(2)) p^(2) -2 (ab + bc + cd) p + (b^(2) + c^(2) + d^(2)) le 0`
`rArr (a^(2) p^(2) - 2abp + b^(2)) + (b^(2) p^(2) -2bcp + c^(2)) + (c^(2) p^(2) -2cdp + d^(2)) le 0`
`rArr (ap - b)^(2) + (bp -c)^(2) + (cp -d)^(2) le 0`
[since, sum of squares is never less than zero]
Since, each of the squares is zero.
`:. (ap = b)^(2) = (bp -c)^(2) + (cp -d)^(2) = 0`
`rArr p = (b)/(a) = (c)/(b) = (d)/(c)`
`:.` a, b, c, d are in GP.
Promotional Banner

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(ab + bc + cd) p + (b^2 + c^2 + d^2) le 0 , then show that a, b, c and d are in GP.

If a, b, c, d are distinct integers in A. P. Such that d=a^2+b^2+c^2 , then a + b + c + d is

If a,b , c are distinct +ve real numbers and a^(2)+b^(2)+c^(2)=1" then "ab+bc+ca is

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

If a , b, c, … are in G.P. then 2a,2b,2c,… are in ….

If a,b, and c are in G.P then a+b,2b and b+ c are in

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

If a,b and c are distinct positive real numbers in A.P, then the roots of the equation ax^(2)+2bx+c=0 are

If a,b and c are non- zero real number then prove that |{:(b^(2)c^(2),,bc,,b+c),(c^(2)a^(2),,ca,,c+a),(a^(2)b^(2),,ab,,a+b):}| =0

If a, b, c, d are in G.P, prove that (a^n + b^n), (b^n + c^n), (c^n + d^n) are in G.P.