Let αand βbe the roots of equation p x^2+q x+r-0, p ≠0 . If p, q, r are in A.P. and (1)/(α)+(1)/

step 1 in this problem sin let p q r be in ab since they are given this means that two q should be equa

Let αand βbe the roots of equation p x^2+q x+r-0, p ≠0 . If...

Let $\alpha$ and $\beta$ be the roots of equation $p x^{2}+q x+r-0, p \neq 0 .$ If $p, q, r$ are in A.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=4$, then the value of $|\alpha-\beta|$ is $|2014|$ (A) $\frac{\sqrt{61}}{9}$ (B) $\frac{2 \sqrt{17}}{9}$ (C) $\frac{\sqrt{34}}{9}$ (D) $\frac{2 \sqrt{13}}{9}$

Let $\alpha$ and $\beta$ be the roots of equation $p x^{2}+q x+r-0, p \neq 0 .$ If $p, q, r$ are in A.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=4$, then the value of $|\alpha-\beta|$ is
$|2014|$
(A) $\frac{\sqrt{61}}{9}$
(B) $\frac{2 \sqrt{17}}{9}$
(C) $\frac{\sqrt{34}}{9}$
(D) $\frac{2 \sqrt{13}}{9}$

Key Concepts

Reciprocal Sum of Roots

The sum of the reciprocals of the roots of a quadratic equation can be expressed as (? + ?)/(??). By applying Vieta’s formulas, this expression can be rewritten in terms of the coefficients of the quadratic, which is useful for establishing a link between given conditions on the roots and the coefficients.

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. In the context of quadratic equations, if the coefficients are in arithmetic progression, they satisfy a relation such as 2q = p + r, which imposes a specific condition on the coefficients of the equation.

Vieta's Formulas

Vieta's formulas relate the roots of a polynomial to its coefficients. For a quadratic equation ax² + bx + c = 0, the sum of the roots is given by -b/a and the product by c/a. These formulas allow one to express relationships involving the roots directly through the coefficients without solving the equation.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation, computed as b² - 4ac, determines the nature and difference of the roots. Specifically, the absolute difference between the roots is given by |? - ?| = (?(b² - 4ac))/|a|. This concept is key to finding the numerical difference between the roots when additional conditions are provided.

Transcript

Please give Ace some feedback