Multiple choice

The necessary and sufficient condition for the equation $ \left( 1-{ a }^{ 2 } \right) { x }^{ 2 }+2ax-1=0$ to have roots lying in the interval $(0,1)$ is

  1. $a > 0$
  2. $a < 0$
  3. $a > 2$

  4. none of these
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C Correct answer
Explanation

For the roots of f(x) = (1-a^2)x^2 + 2ax - 1 to lie in (0,1), the discriminant must be non-negative and the function values at boundaries must satisfy specific conditions. Analysis shows the condition is a > 2.

AI explanation

Let us test a value satisfying a > 2, such as a = 3, in the quadratic equation (1 - a^2)x^2 + 2ax - 1 = 0, which becomes -8x^2 + 6x - 1 = 0. Solving this equation yields roots 0.5 and 0.25, which both correctly lie in the interval (0, 1). Conversely, if we choose a value not satisfying the condition, such as a = 1, the equation becomes a linear equation 2x - 1 = 0 with a single root of 0.5 that fails to provide two distinct roots in the interval. Similarly, testing a negative value like a = -1 yields -2x - 1 = 0 with the root -0.5, which does not lie in the required interval. Testing a value between zero and two, like a = 0.5, results in roots outside the required interval. Therefore, the condition a > 2 is both necessary and sufficient for the roots to lie in the specified interval. The result is a > 2.

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