If three distinct numbers a, b, c are in G.P. | vedclass

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

Question

(A) Since $a, b, c$ are in $G.P.$,we have $b^2 = ac$.The equation $ax^2 + 2bx + c = 0$ can be written as $ax^2 + 2\sqrt{ac}x + c = 0$,which is $(\sqrt

Vedclass · Accepted Answer

$\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in $A.P.$

Vedclass · Answer

(A) Since $a, b, c$ are in $G.P.$,we have $b^2 = ac$.The equation $ax^2 + 2bx + c = 0$ can be written as $ax^2 + 2\sqrt{ac}x + c = 0$,which is $(\sqrt{a}x + \sqrt{c})^2 = 0$.Thus,the root of this equation is $x = -\frac{\sqrt{c}}{\sqrt{a}} = -\frac{b}{a}$.Since this is a common root for $dx^2 + 2ex + f = 0$,we substitute $x = -\frac{b}{a}$ into the second equation:$d(-\frac{b}{a})^2 + 2e(-\frac{b}{a}) + f = 0$$d(\frac{b^2}{a^2}) - \frac{2eb}{a} + f = 0$Multiplying by $a^2$,we get $db^2 - 2eab + fa^2 = 0$.Since $b^2 = ac$,we substitute it:$dac - 2eab + fa^2 = 0$.Dividing the entire equation by $ac$,we get:$\frac{d}{a} - \frac{2e}{b} + \frac{f}{c} = 0$,which implies $\frac{d}{a} + \frac{f}{c} = 2(\frac{e}{b})$.This condition indicates that $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in $A.P.$

If three distinct numbers $a, b, c$ are in $G.P.$ and the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root,then which one of the following statements is correct?

Similar Questions

$11^2 + 12^2 + 13^2 + \dots + 20^2 = $

Let $a, b, c \in R$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy$ for all $x, y \in R$,then $\sum_{n=1}^{10} f(n)$ is equal to:

If $(b+c), (c+a), (a+b)$ are in $H.P.$,then $a^2, b^2, c^2$ are in.......

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms,the three terms now form an $A.P.$ Then the sum of the original three terms of the given $G.P.$ is

If $a, b, c$ are in $H.P.$,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module