With respect to a variable point on the line x + y = 2a, chord of contact of the circle x² + y² = a² is drawn. If it passes through a fixed point F, then which of the following is/are true:

$$\begin{gathered} Given circle x^2+y^2=a^2 \\ and the given line x+y=2a \end{gathered}$$

Point on the line is P (t, 2a – t)
Now equation of chord of Contact at P is $\mathrm{xt}+\mathrm{y}(2\mathrm{a}-\mathrm{t})={\mathrm{a}}^2$
$\Rightarrow (\mathrm{x}-\mathrm{y})\mathrm{t}+2\mathrm{ay}-{\mathrm{a}}^2=0$
$\therefore$This passes through F= $\left(\frac{\mathrm{a}}{2},\frac{\mathrm{a}}{2}\right)$
Now equation of chord whose mid point is F is
$\frac{\mathrm{xa}}{2}+\frac{\mathrm{ya}}{2}=\frac{{\mathrm{a}}^2}{2}$