Assume the centre of circle to be origin and (x₀,y₀) to be the midpoint of the chord.Since it subtends 45⁰ at the circumference,it subtends 90⁰ at the centre.It's eqn. is S₀=S₀₀ where S is x²+y²=r²  that is xx₀+yy₀=x₀²+y₀².By homogenisation,

x²+y²=r²[xx₀+yy₀/x₀²+y₀²]².Equating the sum of the coefficients of x² and y² on both sides,we get 2=r²/x₀²+y₀².hence the ans is r/rt2