If a, b, c, d and p are distinct real numbers | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The given inequality is $(a^2 + b^2 + c^2)p^2 - 2p(ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$.This can be rearranged as:$(a^2 p^2 - 2abp + b^2) + (b

Vedclass · Accepted Answer

$a, b, c, d$ are in $G.P.$

Vedclass · Answer

(D) The given inequality is $(a^2 + b^2 + c^2)p^2 - 2p(ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$.This can be rearranged as:$(a^2 p^2 - 2abp + b^2) + (b^2 p^2 - 2bpc + c^2) + (c^2 p^2 - 2cpd + d^2) \le 0$.This simplifies to:$(ap - b)^2 + (bp - c)^2 + (cp - d)^2 \le 0$.Since the sum of squares of real numbers can only be non-negative,the only way for the sum to be less than or equal to $0$ is if each term is exactly $0$.Therefore,$ap - b = 0$,$bp - c = 0$,and $cp - d = 0$.This implies $p = \frac{b}{a} = \frac{c}{b} = \frac{d}{c}$.This is the definition of a Geometric Progression $(G.P.)$,where $a, b, c, d$ are in $G.P.$ with common ratio $p$.

If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^2 + b^2 + c^2)p^2 - 2p(ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$,then

Similar Questions

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-3 \lambda x+\lambda^{2}=0,$ find $\lambda$ if $\alpha^{2}+\beta^{2}=\frac{7}{4}$

The common root of the equations $x^{2}-7x+10=0$ and $x^{2}-10x+16=0$ is:

Solve the given two equations and select the correct answer from the given options.
$I.$ $2.3x - 20.01 = 0$
$II.$ $2.9y - x = 0$

The number of roots of the quadratic equation $8\sec^2 \theta - 6\sec \theta + 1 = 0$ is

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 + x \sin \theta - 2 \sin \theta = 0$,where $\theta \in (0, \frac{\pi}{2})$,then the value of $\frac{\alpha^{12} + \beta^{12}}{(\alpha^{-12} + \beta^{-12})(\alpha - \beta)^{24}}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module