Question

The length of the sub tangent at \[\left( {2,2} \right)\] to the curve ${x^5} = 2{y^4}$ is
${\text{A}}{\text{.}}$$\dfrac{5}{2}$
${\text{B}}{\text{.}}$$\dfrac{8}{5}$
${\text{C}}{\text{.}}$ $\dfrac{2}{5}$
${\text{D}}{\text{.}}$$\dfrac{5}{8}$

Answer 1

Hint: This question is based on length of subtangent formula to the curve equation in which the length of subtangent formula is $\dfrac{y}{m}$ where m can be written as $m = \dfrac{{dy}}{{dx}}$.

Complete step-by-step answer:

As Given curve equation is $2{y^4} = {x^5}$ So first we will do differentiating both sides, such that
$
  8{y^3}\dfrac{{dy}}{{dx}} = 5{x^4} \\
  {\left( {\dfrac{{dy}}{{dx}}} \right)_{2,2}} = \dfrac{{5{{\left( 2 \right)}^4}}}{{8{{\left( 2 \right)}^3}}} \\
  {\left( {\dfrac{{dy}}{{dx}}} \right)_{2,2}} = \dfrac{5}{4} \\
$

Therefore, length of sub tangent $ = \dfrac{y}{{\dfrac{{dy}}{{dx}}}} = \dfrac{2}{{\dfrac{5}{4}}} = \dfrac{8}{5}$

hence the required value is $\dfrac{8}{5}$.

So, option B is the correct answer.

Note: In such type of questions firstly we did the differentiation of the given curve equation ${x^5} = 2{y^4}$ after that we put the value of points $\left( {2,2} \right)$ and found the value of $\dfrac{{dy}}{{dx}}$, then putting this value in the length of sub tangent formula $\dfrac{y}{m} = \dfrac{y}{{\dfrac{{dy}}{{dx}}}}$ and finally get the required result.