For a concave lens of focal length f , the relation between object...

From the lens formula, $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$

$⇒ v = f - \dfrac{{f}^{2}}{u + f}$

 now, if we plot a graph between $u\text{and}v$

When $u \to \infty ; v = f - \dfrac{{f}^{2}}{\infty} = f$

and for $u \to 0; v = 0$

Given that $u = v$is a reference line, therefore, $\dfrac{d v}{d u} = 0 + \dfrac{{f}^{2}}{{\left(u + f\right)}^{2}}$

${\dfrac{d v}{d u}}_{u = 0} = 0 + \dfrac{{f}^{2}}{{\left(0 + f\right)}^{2}} = 1$

The equation of tangent of the curve at the origin, $v - 0 = 1\left(u - 0\right) ⇒ v = u$