Question

The ends $A$ and $B$ of a rod of length $\sqrt 5$ are sliding along the curve $y = 2{x^2}$ . Let ${x_A}$ and ${x_B}$​ be the $x$-coordinate of the ends. At the moment when $A$ is at $\left( {0,0} \right)$ and $B$ is at $\left( {1,2} \right)$ the derivative $\dfrac{{d{x_A}}}{{d{x_B}}}$ has the value equal to
A.$\dfrac{1}{3}$
B.$\dfrac{1}{5}$
C.$\dfrac{1}{8}$
D.$\dfrac{1}{9}$

Answer 1

Here we will assume the coordinates of the end points of the rod to be any variable. We will substitute the coordinates of these points in the equation of the curve and then find the distance between these points by applying the distance formula. Then we will differentiate the given expression with respect to the variable to find the required answer.

Formula Used:
We will use the distance formula which is given by $d = \sqrt {{{\left( {x - {x_1}} \right)}^2} + {{\left( {y - {y_1}} \right)}^2}}$.

Complete step-by-step answer:
It is given that the ends of the rod are sliding along the given curve.
Let the point $A$ of the rod on the curve be $\left( {{x_A},{y_A}} \right)$ and this point will satisfy the equation of the curve.
Now, we will substitute the coordinates of point $A$ in the given equation $y = 2{x^2}$.
${y_A} = 2{x_A}^2$ …………. $\left( 1 \right)$
Let the point $B$ of the rod on the curve be $\left( {{x_B},{y_B}} \right)$ and this point will satisfy the equation of the curve.
Substituting the coordinates of point $B$ in the given equation $y = 2{x^2}$.
${y_B} = 2{x_B}^2$ …………. $\left( 2 \right)$
Now, the distance between the points $A$ and $B$ of a rod will give the length of the rod.
We will find the distance between the two points.
Substituting $x = {x_B},{x_1} = {x_A},y = {y_B}$ and ${y_1} = {y_A}$ in the formula $d = \sqrt {{{\left( {x - {x_1}} \right)}^2} + {{\left( {y - {y_1}} \right)}^2}}$, we get
$\sqrt {{{\left( {{x_B} - {x_A}} \right)}^2} + {{\left( {{y_B} - {y_A}} \right)}^2}} = \left( {\sqrt 5 } \right)$
${\left( {{x_B} - {x_A}} \right)^2} + {\left( {{y_B} - {y_A}} \right)^2} = {\left( {\sqrt 5 } \right)^2}$
Now, we will substitute the value of ${y_A}$ and ${y_B}$ from equation $\left( 1 \right)$ and equation $\left( 2 \right)$.
$\Rightarrow {\left( {{x_B} - {x_A}} \right)^2} + 4{\left( {{x_B}^2 - {x_A}^2} \right)^2} = 5$
Now, we will differentiate both sides with respect to ${x_A}$.
$\Rightarrow - 2\left( {{x_B} - {x_A}} \right)\left( {1 - \dfrac{{d{x_B}}}{{d{x_A}}}} \right) + 8\left( {{x_B}^2 - {x_A}^2} \right)\left( {2\dfrac{{d{x_B}}}{{d{x_A}}} - 2{x_A}} \right) = 0$
As the points $A$ and $B$ are known i.e. $A\left( {0,0} \right)$ and $B\left( {1,2} \right)$.
$\begin{array}{l}{x_B} = 1\\\\{x_A} = 0\end{array}$
On substituting these values here, we get
$\Rightarrow - 2\left( {1 - 0} \right)\left( {1 - \dfrac{{d{x_B}}}{{d{x_A}}}} \right) + 8\left( {{1^2} - {0^2}} \right)\left( {2\dfrac{{d{x_B}}}{{d{x_A}}} - 2 \times 0} \right) = 0$
On further simplification, we get
$\Rightarrow - 2 + 2\dfrac{{d{x_B}}}{{d{x_A}}} + 16\dfrac{{d{x_B}}}{{d{x_A}}} = 0$
On adding and subtracting the like terms, we get
$\Rightarrow 18\dfrac{{d{x_B}}}{{d{x_A}}} = 2$
Dividing both sides by 18, we get
$\Rightarrow \dfrac{{d{x_B}}}{{d{x_A}}} = \dfrac{2}{{18}} = \dfrac{1}{9}$
Therefore, the correct option is option D.

Note: Here we have differentiated the expression with respect to ${x_A}$. We have used the chain rule of differentiation. We have been provided $x$- coordinate, so we will assume $y$- coordinate with similar notation and then substitute in the given equation of the curve. The length of the rod is the distance between the two points because the points are located at the extreme ends of the rod.