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(D) Let the point $P$ be $\left( \frac{1 + \sqrt{2} a}{2}, \frac{1 - \sqrt{2} a}{2} \right)$. Note that $P$ lies on the circle $2x^2 + 2y^2 - (1 + \sq

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$a \in (-\infty, -2) \cup (2, \infty)$

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(D) Let the point $P$ be $\left( \frac{1 + \sqrt{2} a}{2}, \frac{1 - \sqrt{2} a}{2} \right)$. Note that $P$ lies on the circle $2x^2 + 2y^2 - (1 + \sqrt{2} a)x - (1 - \sqrt{2} a)y = 0$. Let the line $x + y = 0$ bisect a chord $PQ$ at point $M(h, -h)$. Since $M$ is the midpoint of $PQ$,the coordinates of $Q$ are $(2h - x_P, -2h - y_P)$. Substituting $Q$ into the circle equation and simplifying,we get a quadratic equation in $h$: $8h^2 - 6\sqrt{2}ah + 1 + 2a^2 = 0$. For the chords to be distinct,the discriminant $\Delta$ of this quadratic must be greater than $0$. $\Delta = (-6\sqrt{2}a)^2 - 4(8)(1 + 2a^2) > 0$ $72a^2 - 32 - 64a^2 > 0$ $8a^2 - 32 > 0$ $a^2 - 4 > 0$ $(a - 2)(a + 2) > 0$ Thus,$a \in (-\infty, -2) \cup (2, \infty)$.

The interval of the values of $a$ for which the line $x + y = 0$ bisects $2$ distinct chords drawn from a point $P \left( \frac{1 + \sqrt{2} a}{2}, \frac{1 - \sqrt{2} a}{2} \right)$ to the circle $2x^2 + 2y^2 - (1 + \sqrt{2} a)x - (1 - \sqrt{2} a)y = 0$ is:

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