Question

Draw the graph of $xy=20$ where $x,y>0$. Use the graph to find $y_1$ when $x=5$ , and to find $x_1$ when $y=10$ . Then find $x_1+y_1$ .$$$$

Answer 1

Hint: We see that the given equation of curve is in implicit form $xy=20$ we write it in the explicit form of function $y=f\left(x\right)={\displaystyle \frac{20}{x}}$. We assume 3 positive values and 3 negative values for $x$. We find the corresponding functional values. $y=f\left(x\right)$ We find the coordinates of points $(x,f\left(x\right))$ and join them to get the plot of the curve . We draw the lines $x=5$ and the ordinate of its point of intersection with the curve $xy=20$ will be $y_1$. We draw the lines $y=10$ and abscissa of its point of intersection with the curve $xy=20$ will be $x_1$. $$$$

Complete step-by-step answer:
We have the given equation of the curve in the question as
$$xy=20$$
Here the condition is the non-negative constraints $x,y>0$ which means the curve is only in first quadrant. We see that it is in implicit form which means it is not expressed in the form $y=f\left(x\right)$. We write explicitly in the form $y=f\left(x\right)$ as
$$\begin{array}{rl}& xy=20\\ & \Rightarrow y={\displaystyle \frac{20}{x}}\\ & \Rightarrow y=f\left(x\right)={\displaystyle \frac{20}{x}}....\left(1\right)\end{array}$$
We can see that above function is clearly not defined for $x=0$ We see that $x$ and $y$ are multiplied to each other and the product is 20. We can take the factors of 20 as assumed integral values of $x$. We have the factors of 20 as 1,2,4,5,10,20. We take 5 values $x=1,2.5,4,8,20$ in equation and find the corresponding values of $y=f\left(x\right)={\displaystyle \frac{20}{x}}$. We write the obtained values in the following table. $$$$
 

We plot the obtained points $(x,f\left(x\right))$ and join the points to obtain the plot of the curve $xy=20$ with $x,y>0$$$$$

 

We are also asked to use the graph to find $y_1$ when $x=5$ , and to find $x_1$ when $y=10$ and then find $x_1+y_1$ . So we draw the lines $x=5$ and $y=10$ and then observe only the first quadrant as there are only positive values and see where the lines intersect the curve. $$$$

We observe that the line $x=4$ intersects the curve $xy=20$ at $(5,4)$, so when $x=4$ we have $y_1=4$. We also observe that the line $y=10$ intersects the curve at $(2,10)$, so when $y=10$ we have $x_1=2$. The required value from the question is $x_1+y_1=2+4=6$.$$$$

Note: We can verify the points of intersection by putting them in the equation of the curve. The plot we obtained is the plot of a rectangular hyperbola whose standard equation is given by $XY=C$ for some real constant $C$. The rectangular hyperbola is special type of hyperbola obtained when we put the condition $a=b$ in the general equation of an hyperbola ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$.