The quadratic equations `x^2""-6x""+""a""=""0""a n d ""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

Options

A. 3
B. 2
C. 1
D. 4

Answer

B

Explanation

Let `alpha` be the common root and other roots be `4 beta and 3 beta` respectively.
`therefore alpha + 4 beta = 6 and 4 alpha beta = a" "[because alpha, 4 beta "are roots of" x^(2) - 6x + a = 0]`
`alpha + 3 beta = c and 3 alpha beta = 6" "[because alpha, 3 beta "are roots of" x^(2) -cx + 6 = 0]`
Now, `4 alpha beta = a and 3 alpha beta = 6 rArr a = 8`
Putting `a = 8 "in" x^(2) - 6x + a = 0`, we get x = 2, 4.
If `alpha = 2 and 4 beta = 4`, then `beta = 1`.
So, roots of the two equations are 2, 4 and 2, 3.
Clearly, 2 is the common root.
If `alpha = 4, 4 beta = 2`, then `beta=(1)/(2)`.
In this case, roots of the second equation are not integer.
So, `alpha = 4, 4 beta = 2` is not possible.