The value of 'a' such that the sum of the squares of the roots of the equation $x^{2}-(a-2)x-a-1=0$ assumes least value, is:

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Accepted Answer

To determine the value of 'a' such that the sum of the squares of the roots of the equation $x^2 - (a-2)x - a - 1 = 0$ assumes the least value, we need to analyze the properties of the roots of the quadratic equation.

The given quadratic equation is:

$$ x^2 - (a-2)x - (a+1) = 0. $$

For a quadratic equation of the form $x^2 + bx + c = 0$, the sum of the roots ($S$) is given by $-b$, and the product of the roots ($P$) is given by $c$. Therefore, for our equation:

- The sum of the roots, $S = a - 2$.
- The product of the roots, $P = -(a + 1)$.

The sum of the squares of the roots can be expressed in terms of $S$ and $P$ as follows:

$$ \text{Sum of squares of roots} = S^2 - 2P. $$

Substituting the expressions for $S$ and $P$:

$$ \text{Sum of squares of roots} = (a - 2)^2 - 2(-a - 1). $$

Simplifying this expression:

$$ = (a - 2)^2 + 2a + 2. $$

$$ = a^2 - 4a + 4 + 2a + 2. $$

$$ = a^2 - 2a + 6. $$

To find the value of 'a' that minimizes this expression, we need to find the vertex of the quadratic function $a^2 - 2a + 6$. The vertex form of a quadratic function $ax^2 + bx + c$ is minimized at $x = -\frac{b}{2a}$.

For our quadratic expression $a^2 - 2a + 6$:

- $a = 1$,
- $b = -2$.

The value of 'a' that minimizes the expression is:

$$ a = -\frac{-2}{2 \times 1} = 1. $$

Thus, the value of 'a' that makes the sum of the squares of the roots assume the least value is indeed 1, which corresponds to Option 4.

**Explanation for Other Options:**

1. **Option 1 (a = 0):** Substituting $a = 0$ into the expression $a^2 - 2a + 6$ gives $0^2 - 2(0) + 6 = 6$.

2. **Option 2 (a = -1):** Substituting $a = -1$ gives $(-1)^2 - 2(-1) + 6 = 1 + 2 + 6 = 9$.

3. **Option 3 (a = 2):** Substituting $a = 2$ gives $2^2 - 2(2) + 6 = 4 - 4 + 6 = 6$.

In all these cases, the value of the expression is greater than when $a = 1$, which yields the minimum value of 5. Therefore, Option 4 is the correct answer.