Theory of Equations 1 Question 19

20. Let α and β be the roots of equation px2+qx+r=0, p0. If p,q and r are in AP and 1α+1β=4, then the value of |αβ| is

(2014 Main)

(a) 619

(b) 2179

(c) 349

(d) 2139

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Solution:

  1. PLAN If ax2+bx+c=0 has roots α and β, then α+β=b/a and αβ=ca. Find the values of α+β and αβ and then put in (αβ)2=(α+β)24αβ to get required value.

Given, α and β are roots of px2+qx+r=0,p0.

α+β=qp,αβ=rp

Since, p,q and r are in AP.

2q=p+r

Also, 1α+1β=4α+βαβ=4

α+β=4αβq=4r

[from Eq. (i)]

On putting the value of q in Eq. (ii), we get

2(4r)=p+rp=9r

 Now, α+β=qp=4rp=4r9r=49 and αβ=rp=r9r=19(αβ)2=(α+β)24αβ=1681+49=16+3681(αβ)2=5281|αβ|=2913



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