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Problem 679

# Write an equation of the locus of points equidistant from the points $$\mathrm{P}_{1}(2,2)$$ and $$\mathrm{P}_{2}(6,2)$$.

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The equation of the locus of points equidistant from the points $$\mathrm{P}_{1}(2,2)$$ and $$\mathrm{P}_{2}(6,2)$$ is $$x=4$$.
See the step by step solution

## Step 1: Find the midpoint of the line segment joining $$\mathrm{P}_{1}$$ with $$\mathrm{P}_{2}$$

The midpoint of a line segment with endpoints $$(x_1,y_1)$$ and $$(x_2,y_2)$$ is given by the formula $$M(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$. Applying this formula to the given points: $$M(\frac{2+6}{2},\frac{2+2}{2}) = M(4,2)$$ So the midpoint of the line segment joining $$\mathrm{P}_{1}$$ and $$\mathrm{P}_{2}$$ is $$M(4,2)$$.

## Step 2: Find the slope of the line segment joining $$\mathrm{P}_{1}$$ with $$\mathrm{P}_{2}$$

The slope of a line segment with endpoints $$(x_1,y_1)$$ and $$(x_2,y_2)$$ can be calculated using the formula $$m = \frac{y_2-y_1}{x_2-x_1}$$. Applying this formula to the given points: $$m = \frac{2-2}{6-2} = \frac{0}{4} = 0$$ The slope of the line segment joining $$\mathrm{P}_{1}$$ and $$\mathrm{P}_{2}$$ is $$0$$.

## Step 3: Determine the slope of the perpendicular bisector

The slope of the perpendicular bisector will be the negative reciprocal of the slope of the line segment joining the points $$\mathrm{P}_{1}$$ and $$\mathrm{P}_{2}$$. Since the slope of the line segment is $$0$$, the perpendicular bisector will have a slope that is undefined or a vertical line.

## Step 4: Use the point-slope form to find the equation of the perpendicular bisector

Since the slope of the perpendicular bisector is undefined, this implies that the perpendicular bisector is a vertical line. A vertical line has a constant $$x$$ value. Since the midpoint of the line segment joining $$\mathrm{P}_{1}$$ and $$\mathrm{P}_{2}$$ is $$M(4,2)$$, the equation of the perpendicular bisector will be: $$x=4$$ So, the equation of the locus of points equidistant from the points $$\mathrm{P}_{1}(2,2)$$ and $$\mathrm{P}_{2}(6,2)$$ is $$x=4$$.

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