Question

Why all the n-o bonds in nitrate ion are of equal length?

Answer 1

All N-O bonds in NO₃⁻ are the same length.

You can draw three equivalent Lewis structures for NO₃⁻. Each has an N=O double bond and two N-O single bonds.

The actual structure is none of these. It is a resonance hybrid of them all.

All three bonds have exactly the same length.