Question

If the length of sub-normal is equal to the length of subtangent at any point $(3,4)$ on the curve $y=f(x)$ and the tangent at $(3,4)$ to $y=f(x)$ meets the coordinate axes at $A$ and $B$, then the maximum area of the triangle $OAB$, where $O$ is origin, is

• A. 45/2
• B. 49/2
• C. 25/2
• D. 81/2

Answer 1

Answer: (B)

Length of sub-normal = length of sub-tangent or $\frac{dy}{dx}=\pm 1$
If $\frac{dy}{dx}=1$, equation of the tangent is
$y-4=x-3$ or $y-x=1$
Area of $\Delta OAB=\frac{1}{2} \times 1 \times 1=\frac{1}{2}$
If $\frac{dy}{dx}=-1$, equation of the tangent is:
$ y-4=-x+3$ or $ y+x=7$,
$\therefore $ Area $=\frac{1}{2} \times 7 \times 7=\frac{49}{2}$