If the sum of the roots of the quadratic equa | vedclass

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

Question

(C) Let $\alpha$ and $\beta$ be the roots of the quadratic equation $ax^2 + bx + c = 0$. Then,$\alpha + \beta = -b/a$ and $\alpha\beta = c/a$. Accordi

Vedclass · Accepted Answer

$H.P.$

Vedclass · Answer

(C) Let $\alpha$ and $\beta$ be the roots of the quadratic equation $ax^2 + bx + c = 0$. Then,$\alpha + \beta = -b/a$ and $\alpha\beta = c/a$. According to the problem,$\alpha + \beta = \frac{1}{\alpha^2} + \frac{1}{\beta^2}$. This implies $\alpha + \beta = \frac{\alpha^2 + \beta^2}{(\alpha\beta)^2} = \frac{(\alpha + \beta)^2 - 2\alpha\beta}{(\alpha\beta)^2}$. Substituting the values,we get $-b/a = \frac{(-b/a)^2 - 2(c/a)}{(c/a)^2} = \frac{b^2/a^2 - 2c/a}{c^2/a^2} = \frac{b^2 - 2ac}{c^2}$. Thus,$-bc^2 = a(b^2 - 2ac) = ab^2 - 2a^2c$. Rearranging,$2a^2c = ab^2 + bc^2$. Dividing both sides by $abc$,we get $\frac{2a}{b} = \frac{b}{c} + \frac{c}{a}$. This shows that $\frac{c}{a}, \frac{a}{b}, \frac{b}{c}$ are in $A.P.$ Therefore,their reciprocals $\frac{a}{c}, \frac{b}{a}, \frac{c}{b}$ are in $H.P.$

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals,then $a/c, b/a, c/b$ are in

Similar Questions

If $\alpha, \beta$ are the roots of the equation $x^2+3x+k=0$ and $\alpha+\frac{1}{\alpha}, \beta+\frac{1}{\beta}$ are the roots of the equation $4x^2+px+18=0$,then $k$ satisfies the equation:

Let $f(x)$ be a quadratic polynomial such that $f(-2) + f(3) = 0$. If one of the roots of $f(x) = 0$ is $-1$,then the sum of the roots of $f(x) = 0$ is equal to

If the harmonic mean between the roots of $(5+\sqrt{2}) x^2-b x+(8+2 \sqrt{5})=0$ is $4$,then the value of $b$ is

The sum of the roots of a quadratic equation is $2$ and the sum of their cubes is $98$. Then the equation is:

If the ratio of the roots of the equation $px^2+qx+r=0$ is $a:b$, then $\frac{ab}{(a+b)^2}=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module