Question

Consider the point $(x,y)$ lying on the graph of the line line .$2x+4y=5.$ Let $L$ be the distance from the point $(x,y)$ to the origin $(0,0)$ Write $L$ as a function of $x$.

Answer 1

Writing $L$ as a function of $x$.

Given equation of the line is ,

$$\begin{gathered} 2x+4y=5 \\ \Rightarrow y=\frac{-1}{2}x+\frac{5}{4}......\left(i\right) \end{gathered}$$

Finding $L$ in terms of $x$,

$$\begin{gathered} L=\sqrt{{(x-0)}^2+{(y-0)}^2} \\ =\sqrt{x^2+{\left(\frac{-1}{2}x+\frac{5}{4}\right)}^2} \\ =\sqrt{x^2+\frac{1}{4}{x}^2-\frac{5}{4}x+\frac{25}{16}} \\ =\sqrt{\frac{5}{4}{x}^2-\frac{5}{4}x+\frac{25}{16}} \\ =\sqrt{\frac{20x^2-20x+25}{16}} \\ =\frac{\sqrt{20x^2-20x+25}}{4} \end{gathered}$$

Hence, the value of $L$ in terms of $x$ is $L=\frac{\sqrt{20x^2-20x+25}}{4}$.