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Q.
The set of all values of $a^2$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $P \left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2 x^2+2 y^2-(1+a) x-(1-a) y=0$, is equal to :
$ x ^2+ y ^2-\frac{(1+ a ) x }{2}-\frac{(1- a ) y }{2}=0 $
$ \text { Centre }\left(\frac{1+ a }{4}, \frac{1- a }{4}\right) \Rightarrow( h , k )$
$P \left(\frac{1+ a }{2}, \frac{1- a }{2}\right) \Rightarrow(2 h , 2 k )$
Equation of chord $\Rightarrow T=S_1$
$ \Rightarrow( x - y ) \lambda-\frac{2 h ( x +\lambda)}{2}-\frac{(2 k )( y -\lambda)}{2}$
$ =2 \lambda^2-2 h (\lambda)+2 k \lambda$
Now, $\lambda(2 h , 2 k )$ satisfies the chord
$ \therefore(2 h -2 k ) \lambda- h ( x +\lambda)- k ( y -\lambda) $
$ \Rightarrow 2 \lambda^2+4 k \lambda-4 h \lambda+ h \lambda- k \lambda+ hx + ky =0 $
$ \Rightarrow 2 \lambda^2+\lambda(3 k -3 h )+ ky + hx =0 $
$ \Rightarrow D >0 $
$ \Rightarrow 9( k - h )^2-8( ky + hx )>0 $
$ \Rightarrow 9( k - h )^2-8\left(2 k ^2+2 h ^2\right)>0 $
$ \Rightarrow-7 k ^2-7 h ^2-18 kh >0 $
$ \Rightarrow 7 k ^2+7 h ^2+18 kh <0 $
$ \Rightarrow 7\left(\frac{1- a }{4}\right)^2+7\left(\frac{1+ a }{4}\right)^2+18\left(\frac{1- a ^2}{16}\right)<0$
$ \Rightarrow 7\left[\frac{2\left(1+ a ^2\right)}{16}\right]+\frac{18\left(1- a ^2\right)}{16}<0, a ^2= t$
$ \Rightarrow \frac{7}{8}(1+ t )+\frac{18(1- t )}{16}<0 $
$\Rightarrow \frac{14+14 t +18-18 t }{16}<0 $
$ \Rightarrow 4 t >32$
$ t >8 \,\,\,\,a ^2>8$
