Question

if the ratio of roots of the equation lx^2+nx+n=0 is p:q then prove that √p/q + √q/p +√n/l =0

Answer 1

To solve the question, proceed in the following manner- 

Let a and b be those two roots of the given equations. 

√(p/q) + √(q/p) + √(n/l) can be written as,  √p/√q + √q/√p + √n/√l

= (√p² + √q² )/(√p*√q) + √n/√l

= (p + q)/(√p*√q) + √n/√l ...................(1)

Given that a : b = p : q

Let a = px, b = qx

a + b = -n/l

⇒ px + qx = -n/l

⇒ (p + q)x = -n/l

⇒p + q = -n/lx .............(2)

Also a*b = n/l

⇒ px * qx = n/l

⇒ pq*x² = n/l

⇒ √pq * x = √(n/l)

⇒ √pq = √(n/l)/x ..........(3)

Now from equation 1, 

    (-n/lx)/(√(n/l)/x) + √(n/l)

= (-n/l)/(√(n/l)) + √(n/l)

= -√(n/l) + √(n/l)

= 0

So, √(p/q) + √(q/p) + √(n/l) = 0

Hence Proved. 

Answer 2

Hope it helps you all