Help please on locus

[baseball]

(1) How many points are there in a plane that are 4 cm. from a given line and also 5 cm. from a given point on that line?

(2) How many points are there that are equidistant from two given points A and B and also 2 inches from the line passing through A and B?

(3) Given <ABC, what is the number of points found in the interior of the angle, equidistant form the sides of the angle, and also equidistant from points A and B?

(4) The number of points equidistant from two given parallel lines and at the same time equidistant from two given points on oneof these line is

(A) 1 (C) 3
(B) 2 (D) 4

(5) Line AB and CD are paralllel and are 6 inches apart. Point P is on line AB. The number of points equidistant from these two lines and also 5 inches from point P is

(A)1 (C)0
(B)2 (D)4

(6) The triangle whose vertices are A(1,2), B(7,2), and C(5,6).

(PART A) From the locus points equidistant from A and B, write an equation of the locus.
(PART B) Write an equation of the line passing through A and C.
(PART C) Find the coordinates of point P if P is a point on line AC that is equidistant from points A and B.

[minime1235813]

For number 1, There are either 2, 1, or 0 because if the points are far apart or very close apart, as illustrated, the lines would not connect. But if they are the right distance they would connect twice. They could also touch at one point.

Hope thats clear.
(2) needs to be clarified. When it says "and also 2 inches from the line passing through A and B" , that doesnt make sense. What part?
(3)Still doesnt make Sense. There are infinite. Its a line
(4) one

[fishythefish]

For #1, there are 4. They lie where the circle of radius 5 and the center at the given point intersect the two lines parallel to the first one, either 4 inches above or below.

For #2, there are 2 points. They both lie on the perpendicular bisector, and there is one two inches away on either side of AB.

For #3, there is only one. It lies where the perpendicular bisector of AB intersects the angle bisector.

For #4, the answer should be (A) 1. The point lies where the perpendicular bisector of the segment connecting the two points intersects the line parallel to and halfway between the other two lines.

For #5, the answer is (B) 2. The points lie where the circle with center P and radius 5 intersects the line parallel to and halfway between AB and CD.

For #6A, the locus is the perpendicular bisector of AB. The solution is easy because the y-coordinates are the same.
For #6B, let the slope be .



Using point-slope form:





For #6C, P is located at the intersection of the perpendicular bisector of AB and the line AC. We have the two equations from parts A and B:




Therefore,



So the solution is