Question

If the length of subnormal is equal to the length of sub tangent at any point (3,4) on the curve y=f(x) and the tangent at (3,4) to y=f(x) meets the coordinate axis at A and B, the maximum area of $$△OAB.$$ where O is origin, is

Answer 1

Hint: Important formulas used to solve such questions:
Length of subtangent- $$y\times {\displaystyle \frac{dx}{dy}}$$
Length of subnormal- $$y\times {\displaystyle \frac{dy}{dx}}$$
Equation of tangent passing through the point (a,b) and having slope m is given by- $$(y-a)=m(x-b)$$

Complete step-by-step answer:

Given: Length of subtangent= length of subnormal
$$\begin{gathered} \Rightarrow y\times {\displaystyle \frac{dx}{dy}}=y\times {\displaystyle \frac{dy}{dx}} \\ \Rightarrow {\left({\displaystyle \frac{dy}{dx}}\right)}^2=1 \\ \Rightarrow {\displaystyle \frac{dy}{dx}}=\pm 1 \end{gathered}$$
Now, Equation of tangent passing through point (3,4) and having slope 1 is given as:
$$\begin{gathered} \Rightarrow (y-a)=m(x-b) \\ \Rightarrow (y-4)=1(x-3) \\ \Rightarrow y=x+1 \end{gathered}$$
So, the coordinates of A and B are (-1,0) and (0,1) respectively.
Area of $$\mathrm{△}OAB$$is given as:
$$\begin{gathered} \Rightarrow {\displaystyle \frac{1}{2}}\times OA\times OB \\ \Rightarrow {\displaystyle \frac{1}{2}}\times 1\times 1......(O=origin) \\ \Rightarrow {\displaystyle \frac{1}{2}}sq.units \end{gathered}$$
Similarly, equation of tangent passing through point (3,4) and having slope 1 is given as:
$$\begin{gathered} \Rightarrow (y-a)=m(x-b) \\ \Rightarrow (y-4)=(-1)(x-3) \\ \Rightarrow y-4=3-x \\ \Rightarrow y+x=7 \end{gathered}$$
So, the coordinates of A and B are (7,0) and (0,7) respectively.
Area of $$\mathrm{△}OAB$$is given as:
$$\begin{gathered} \Rightarrow {\displaystyle \frac{1}{2}}\times OA\times OB \\ \Rightarrow {\displaystyle \frac{1}{2}}\times 7\times 7......(O=origin) \\ \Rightarrow {\displaystyle \frac{49}{2}}sq.units \end{gathered}$$
So, the maximum area of $$\mathrm{△}OAB$$ is $${\displaystyle \frac{49}{2}}sq.units$$.

Note: In above question we have given that the tangent meets coordinate axes means it will meet at abscissa and ordinate axes. The above coordinates can be found by putting once x=0 and then y=0.