The range of value of $a$ for which the line $y+x=0$ bisects two chords drawn from a point $\displaystyle \left( \frac { 1+\sqrt { 2 } a }{ 2 } ,\frac { 1-\sqrt { 2 } a }{ 2 } \right) $ to the circle $2{ x }^{ 2 }+2{ y }^{ 2 }-\left( 1+\sqrt { 2 } a \right) x-\left( 1-\sqrt { 2 } a \right) y=0$ is

The center of the circle 2x squared plus 2y squared minus (1 plus square root 2 times a)x minus (1 minus square root 2 times a)y = 0 is ((1 plus square root 2 times a) divided by 4, (1 minus square root 2 times a) divided by 4). The image of the given point ((1 plus square root 2 times a)/2, (1 minus square root 2 times a)/2) across the line y = negative x is (negative (1 minus square root 2 times a)/2, negative (1 plus square root 2 times a)/2). For the line to bisect the chords, this image point must lie inside the circle, requiring its distance from the center to be less than the radius. Solving the resulting inequality 4a squared greater than 16 yields a less than negative 2 or a greater than 2.

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