If the length of the subnormal is equal to length of the subtangent at any point (3,4) on the curve $y=f\left(x\right)$ and the tangent at (3, 4) to  $y=f\left(x\right)$  meets the coordinate axes at A and B then maximum area of $\Delta$OAB, where O is origin is.

Length of the subnormal = length of the subtangent  $\frac{dy}{dx}=\pm 1\Rightarrow m=\pm 1$ If  m=1, equation of the tangent is $y-4=x-3\Rightarrow y-x=1$

Area of  $\Delta OAB=\frac{1}{2}\times 1\times 1=\frac{1}{2}$

If m=-1, equation of the tangent is  $y-4=-1\left(x-3\right)\Rightarrow x+y=7$

Area of the  $\Delta AOB=\frac{1}{2}\times 7\times 7=\frac{49}{2}$

Maximum area  =  $\frac{49}{2}$