Question

28. Ifp,q, r are in GP, and the equation px^2+ 2qx + r = 0 and dx^2 + 2ex +f = 0 have a common root,

then show that d/p, e/q, f/r
are in AP

please explain for 6 marks

WHO WILL ANSWER FIRST WILL BE MARKED AS BRAINLIEST BY ME​

Answer 1

Answer:

it is given that,

p,q,r  in  G.P.

So, thier common ratio is same

pq=qr

q2=rp−−−(1)

solving the equation

px2+2qx+r=0

compare that

ax2+bx+c=0

roots are

x=2a−b±b2−4ac

here, a=ϕ,b=2q  andc=r

x=2p−2q±4q2−4pr

x=2p−2q±4pr−4pr

x=2p−2q±0

x=−pq

Thus,

x=−pq is the root of the equation 

px2+2qx+r=0

Also, given that equations px2+2qx+r=0

and dx2+2ex+f=0 have a common root

So, −pq is a root of dx2+2ex+f=0 

putting x=−pq  in dx2+2ex−f=0

d(p−q)2+2e(p−q)+f=0

p2dq2−p2eq+f=0

dq2−2eqp+fp2=0−−−(2)

But We need to show that

pd,qe,rf are in an A.P. 

Now, from equation(2) and weget.

dq2−2eqp+fp2=0

On dividing this pq2

dq2−pq22epq+pq2fp2=pq20

pd−q2e+q2fp=0

pd+q2fp=q2e

putting q2=pr

dp+prfp=q2e

pd+rf=q2e

∴pd,qe,rf, are in an A.P.

Hence proved.

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