If the length of subnormal is equal to length of sub-tangent at 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 maximum

Question

If the length of subnormal is equal to length of sub-tangent at 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 maximum area of the △ O A B where O is origin, is

TestHub · Accepted Answer

Length of subnormal = length of subtangent $\Rightarrow y_{1} (\frac{d y}{d x})_{(x_{1}, y_{1})} = \frac{y _{1}}{( \frac{d y}{d x} ) _{(x_{1}, y_{1})}} \Rightarrow (\frac{d y}{d x})_{(3, 4)} = \pm 1$

If $(\frac{d y}{d x})_{(3, 4)} = 1$ : tangent is $y - x = 1$ and area of $△ O A B = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}$ If $(\frac{d y}{d x})_{(3, 4)} = - 1 :$ tangent is $x + y = 7$ and area of $△ O A B = \frac{1}{2} \times 7 \times 7 = \frac{49}{2}$ ∴ maximum area $= \frac{49}{2}$ sq.units

Mathematics - Application of Derivative Question with Solution | TestHub

Options:

Doubts & Discussion