JEE Main · 2008 · 27 Apr 2008 · Offline · Quadratic Equations and Inequalities

The quadratic equations

x^{2} - 6 x + a = 0

and

x^{2} - c x + 6 = 0

have one root in common. The other roots of the first and second equations are integers in the ratio

4 : 3

. Then the common root is

Answer: (D)

Solution

Let

α

be the shared root and let the other roots be

4 t

and

3 t

for the two equations. From the products,

α \cdot 3 t = 6

gives

α t = 2

, so the first equation's constant is

a = α \cdot 4 t = 8

, making it

x^{2} - 6 x + 8 = 0

with roots

2

and

4

. Testing

α = 2

gives the other roots

4

and

3

, both integers in the ratio

4 : 3

, whereas

α = 4

would force non-integer roots; so the common root is

2

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option