Elasticity of demand is constant and equal to 1 at every point on the demand curve. The shape of such a demand curve would be :

Rectangular hyperbola

When the elasticity of demand is constant and equal to 1 at every point on the demand curve, it implies unitary elasticity i.e. this demand curve has a property that a percentage change in price along the demand curve always leads to equal percentage change in quantity. In such a case, the shape of the demand curve would be a rectangular hyperbola. A rectangular hyperbola has the unique characteristic that the absolute value of elasticity is always equal to 1 along any point on the curve.