# Using analytic geometry, find the coordinates of the point on the x ax

Using analytic geometry, find the coordinates of the point on the x axis which is equidistant from $$\displaystyle{A}{\left({5},\ {8}\right)}$$ and $$\displaystyle{B}{\left(-{3},\ {4}\right)}$$

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Opeance1951

Step 1
Given, $$\displaystyle{A}{\left({5},\ {8}\right)},\ {B}{\left(-{3},\ {4}\right)}$$
Let, equidistent point, $$\displaystyle{p}{\left({x},\ {0}\right)}$$
So, using distance formula:
$$\displaystyle\sqrt{{{\left({x}-{5}\right)}^{{{2}}}+{\left({8}\right)}^{{{2}}}}}=\sqrt{{{\left({x}+{3}\right)}^{{{2}}}+{\left(-{4}\right)}^{{{2}}}}}$$
Both Side:
$$\displaystyle{{{x}^{{{2}}}}}+{25}-{10}{x}+{64}={{{x}^{{{2}}}}}+{9}+{6}{x}+{16}$$
$$\displaystyle{25}+{64}-{9}-{16}={10}{x}+{6}{x}$$
$$\displaystyle{16}{x}={64}$$
$$\displaystyle{x}={4}$$
Coordinate is $$\displaystyle{\left({4},\ {0}\right)}$$