If the sum of the roots of the equation ax^2 | vedclass

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

Question

(A) Given the equation $ax^2 + bx + c = 0$. Let the roots be $\alpha$ and $\beta$. Then $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}

Vedclass · Accepted Answer

$A.P.$

Vedclass · Answer

(A) Given the equation $ax^2 + bx + c = 0$. Let the roots be $\alpha$ and $\beta$. Then $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.The sum of the reciprocals of their squares is $\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{\alpha^2\beta^2} = \frac{(\alpha + \beta)^2 - 2\alpha\beta}{(\alpha\beta)^2}$.Substituting the values: $\frac{(-\frac{b}{a})^2 - 2(\frac{c}{a})}{(\frac{c}{a})^2} = \frac{\frac{b^2}{a^2} - \frac{2c}{a}}{\frac{c^2}{a^2}} = \frac{b^2 - 2ac}{c^2}$.According to the given condition,$\alpha + \beta = \frac{1}{\alpha^2} + \frac{1}{\beta^2}$,so $-\frac{b}{a} = \frac{b^2 - 2ac}{c^2}$.Cross-multiplying gives $-bc^2 = a(b^2 - 2ac) = ab^2 - 2a^2c$.Rearranging the terms,we get $2a^2c = ab^2 + bc^2$.Dividing by $a^2c^2$ is not necessary here; we observe that $2(ca^2) = bc^2 + ab^2$. This is the condition for $bc^2, ca^2, ab^2$ to be in $A.P.$

If the sum of the roots of the equation $ax^2 + bx + c = 0$ is equal to the sum of the reciprocals of their squares,then $bc^2, ca^2, ab^2$ will be in

Similar Questions

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x^{2} + (2 - \lambda)x + (10 - \lambda) = 0$ is minimum,then the magnitude of the difference of the roots of this equation is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-3x^2+x+5=0$,then $y=\Sigma \alpha^2+\alpha \beta \gamma$ satisfies the equation

If $f(x)=ax^2+bx+c$ satisfies $f(1)+2f(2)=0$ and $2f(1)+f(2)=0$,then $3a+b=$

If one real root of the quadratic equation $81x^2 + kx + 256 = 0$ is the cube of the other root,then a value of $k$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $5x^3 - 2x - 4 = 0$,then $\alpha^3 + \beta^3 + \gamma^3 = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module