If a and b are the roots of x² - 3x + p = 0 and c, d are the roots of x² -12x + q = 0 where a, b, c, d form a G.P. Prove that (q + p) : (q - p) = 17 : 15

Solution:

It is given that a and b are the roots of x² - 3x + p = 0

Therefore,

a + b = 3, ab = p (by sum and product of roots of a quadratic equation)....(1)

Also, c and d are the roots of x² - 12x + q = 0

c + d = 12, cd = q ....(2)

It is given that a,b, c, d form a G.P.

Let a = x, b = xr, c = xr², d = xr³

Form (1) and (2) , we obtain

x + xr = 3 ⇒ x (1+ r) = 3 ....(3)

xr² + xr³ = 12 ⇒ xr² (1 + r) = 12 ....(4)

On dividing (4) by (3) , we obtain

[xr² (1+ r)] / [x (1+ r)] = 12/3

⇒ r² = 4

⇒ r = ± 2

Case I:

when r = 2, x = 1

ab = x²r = 2 = p

cd = x²r⁵ = 32 = q

Therefore,

⇒ (q + p)/(q - p) = (32 + 2)/(32 - 2) = 34/30 = 17/15

⇒ (q + p) : (q - p) = 17 : 15

Case II: when r = - 2, x = - 3

ab = x²r = - 18 = p

cd = x²r⁵ = - 288 = q

Therefore,

⇒ (q + p)/(q - p) = (- 288 - 18)/(- 288 + 18) = (- 306)/(- 270) = 17/15

⇒ (q + p) : (q - p) = 17 : 15

Thus, (q + p) : (q - p) = 17 : 15

Hence proved

---

NCERT Solutions Class 11 Maths Chapter 9 Exercise ME Question 18

If a and b are the roots of x² - 3x + p = 0 and c, d are the roots of x² -12x + q = 0 where a, b, c, d form a G.P. Prove that (q + p) : (q - p) = 17 : 15

Summary:

If a and b are the roots of x² - 3x + p = 0 and c, d are the roots of x² -12x + q = 0 where a, b, c, d form a G.P, then we proved that (q + p) : (q - p) = 17 : 15