Question

The point (x,y) lies on the graph of y=x³. Write

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asked 2021-08-11
The point (x,y) lies on the graph of \(\displaystyle{y}={x}³\). Write the distance d between (1, 0) and (x,y) in terms of x. Write the distance d between (0, 1) and (x,y) in terms of x.

Expert Answers (1)

2021-08-12
Use the Distance Formula: \(\displaystyle{d}=\sqrt{{{\left({x}{2}-{x}{1}\right)}^{{2}}+{\left({y}{2}-{y}{1}\right)}^{{2}}}}\)
The distance between (1,0) and (x,y) is: \(\displaystyle{d}{1}=\sqrt{{{\left({x}-{1}\right)}^{{2}}+{\left({y}-{0}\right)}^{{2}}}}\)
\(\displaystyle{d}{1}=\sqrt{{{\left({x}-{1}\right)}^{{2}}+{y}^{{2}}}}\)
Since \(\displaystyle{y}={x}^{{3}}\), we substitute:
\(\displaystyle{d}{1}=\sqrt{{{\left({x}^{{2}}-{2}{x}+{1}\right)}+{\left({x}^{{3}}\right)}^{{2}}}}\)
\(\displaystyle{d}{1}=\sqrt{{{\left({x}^{{2}}-{2}{x}+{1}\right)}+{x}^{{6}}}}\)
\(\displaystyle{d}{1}=\sqrt{{{\left({x}^{{6}}\right)}+{x}^{{2}}-{2}{x}+{1}}}\)
Similarly, the distance between (0,1) and (x,y) is: \(\displaystyle{d}{2}=\sqrt{{{\left({x}-{0}\right)}^{{2}}+{\left({y}-{1}\right)}^{{2}}}}\)
\(\displaystyle{d}{2}=\sqrt{{{\left({x}^{{2}}\right)}+{\left({y}-{1}\right)}^{{2}}}}\)
Since \(\displaystyle{y}={x}^{{3}}\), we substitute: \(\displaystyle{d}{2}=\sqrt{{{\left({x}^{{2}}+{\left({x}^{{3}}-{1}\right)}^{{2}}\right)}}}\)
\(\displaystyle{d}{2}=\sqrt{{{x}^{{2}}+{\left({x}^{{6}}-{2}{x}^{{3}}+{1}\right)}}}\)
\(\displaystyle{d}{2}=\sqrt{{{x}^{{6}}-{2}{x}^{{3}}+{x}^{{2}}+{1}}}\)
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