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Problem 690

Two concentric circles have radii whose lengths are 2 in. and 6 in. Line \(\mathrm{m}\) is drawn, in the accompanying figure, tangent to the smaller circle, (a) Describe fully the locus of points equidistant from the two circles, (b) Describe fully the locus of points at a given distance \(d\) from line \(\mathrm{m}\). (c) How many points are there which satisfy the conditions given in both parts (a) and (b) if: (1) \(\mathrm{d}<2\) in.? (2) \(\mathrm{d}=2\) in.? (3) \(\mathrm{d}=6\) in.? (4) \(\mathrm{d}>6\) in.?

Expert verified

In conclusion:
- For \(d < 2\), there are 2 points satisfying the conditions.
- For \(d = 2\), there is 1 point satisfying the conditions.
- For \(d = 6\), there is 1 point satisfying the conditions.
- For \(d > 6\), there are no points satisfying the conditions.

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Chapter 39

Describe the locus of points determined by the equation \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}=2 \mathrm{x}\). (Hint: Complete the square in x.)

Chapter 39

What is the locus of points traced by the center of a circle with radius \(\mathrm{R}_{2}\) that rolls around a second circle, the radius of which is \(R_{1}\) ?

Chapter 39

Write an equation of the locus of points in which the ordinate of each point is 3 more than 4 times the abscissa of that point.

Chapter 39

Write an equation for the locus of points equidistant from \((3,3)\) and \((4,4)\).

Chapter 39

(a) Describe the locus of points 2 units from the y-axis and write an equation of this locus, (b) Describe the locus of points equidistant from the points \(\mathrm{P}_{1}(-4,2)\) and \(\mathrm{P}_{2}(-4,6)\) and write an equation for this locus. (c) Find the number of points which satisfy both conditions stated in (a) and (b) and give the coordinates of each point.

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