Question

If the length of subnormal is equal to the length of subtangent at a 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 twice the value of maximum area of the triangle $OAB$, where $O$ is origin is:

Answer 1

Given: Length of subnormal $=$ length of the subtangent
$\Rightarrow y\frac{dy}{dx}=y\frac{dx}{dy}$
$\Rightarrow \frac{dy}{dx}=\pm 1$
Equation of the tangent is $y-4=x-3\Rightarrow y-x=1$ or $y-4=-x+3\Rightarrow x+y=7$
$\Rightarrow$
The corresponding intercepts are $1\text{and}1$ or $7\text{and}7$
The area of $\Delta OAB=\frac{1}{2}\times 1\times 1=\frac{1}{2}$
In the other case, area $=\frac{1}{2}\times 7\times 7=\frac{49}{2}$