Yet Another Geometric Question

Rearrange the circle equation we have $:\\$ $(x+1)^2+(y-2)^2=2^2$ --> Centre $(-1, 2)$ , $r = 2$ $\\$ Chord length $= 4 = 2\sqrt {r^2-d^2} --> 4 = 2\sqrt {4-d^2} --> d = 0$ [We realize that the line is a diameter] $\\$ Substitude $(-1, 2)$ into the line : $\\$ $-2a-2b+2=0$ $\\$ $a+b=1$ $\\$ So $:\\$ $\frac 1a + \frac 1b = \frac {a+b}{ab} = \frac {1}{ab} \geq \frac {1}{(\frac {a+b}{2})^2} = 4$