JEE Main 2014 — Mathematics Algebra

$$\frac{ 1 }{ \alpha } + \frac{ 1 }{ \beta } = 4$$α1​+β1​=4

$$\frac{ \alpha + \beta }{ \alpha \beta } = 4$$αβα+β​=4

$$\therefore \frac{-\frac{q}{p}}{\frac{r}{p}}=4$$∴pr​−pq​​=4

$$\therefore -\frac{q}{r}=4$$∴−rq​=4

$p,q,r$p,q,r are in A.P.

Let $p=a-d;q=a,r=a+d$p=a−d;q=a,r=a+d.

$$\therefore \frac{-(a)}{a+d}=4\Rightarrow -a=4a+4d\Rightarrow d=\frac{-5a}{4}$$∴a+d−(a)​=4⇒−a=4a+4d⇒d=4−5a​

$$p=a+\frac{5a}{4}=\frac{9a}{4};q=a,r=a-\frac{5a}{4}=-\frac{a}{4}$$p=a+45a​=49a​;q=a,r=a−45a​=−4a​

$$\therefore \frac{9}{4}{x}^{2}+\frac{4}{4}x-\frac{1}{4}=0$$∴49​x2+44​x−41​=0

$$\therefore 9{x}^{2}+4x-1=0$$∴9x2+4x−1=0

$${ | \alpha - \beta | }^{2} = { ( \alpha + \beta ) }^{2} - 4 \alpha \beta$$∣α−β∣2=(α+β)2−4αβ

$$= { ( \frac{ - 4 }{ 9 } ) }^{2} - 4 ( - \frac{ 1 }{ \text{9} } )$$=(9−4​)2−4(−91​)

$$= \frac{ 1 6 }{ 8 1 } + \frac{ 4 }{ 9 } = \frac{ 5 2 }{ 8 1 }$$=8116​+94​=8152​

$\therefore | \alpha - \beta | = \frac{ 2 \sqrt{ 1 3 } }{ 9 }$∴∣α−β∣=9213​​.

Hence, the correct answer is $\frac{ 2 \sqrt{ 1 3 } }{ 9 }$9213​​.