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If b gt a, then the equation (x-a)(x-b)-...

If `b gt a`, then the equation `(x-a)(x-b)-1=0` has
(a) Both roots in `(a, b)` `" "` (b) Both roots in `(-oo, a)`
(c) Both roots in `(b, +oo)` `" "` (d) One root in `(-oo, a)` and the other in `(b, +oo)`

A

both roots in [a, b]

B

both roots in `(-oo, a]`

C

both roots in `(b, oo)`

D

one roots in `(-oo, a)` and other in `(b, oo)`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `f(x) = (x - a)(x - b) -1`.
We observe that the coefficient of `x^(2)` in f(x) is positive and `f(a) = f(b) = - 1`. Thus, the graph of f(x) is as shown in Fig 18.

It is evident from the graph that one of the roots of f(x) = 0 lies in `(-oo, a)` and the other lies in `(b, oo)`.
Hence, option (d) is correct.
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