Consider the line segment joining A(-1,2) and B(3,4)
a) find an equation that expresses the fact that a point P(x,y) is equidistant from A and from B.
b) describe geometrically the set of points described by the equation in part (a).
a) arriving at the equation only using your definition for P .....
AP = BP
√( (x+1)^2 + (y-2)^2 ) = √( (x-3)^2 + (y-4)^2 )
square both sides and expand
x^2 + 2x + 1 + y^2 -4y + 4 = x^2 - 6x + 9 + y^2 - 8y + 16
8x - 12y = 20 or
2x - 3y = 5
b) of course the equation we found in a) is the right - bisector of AB
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a) To find an equation that expresses the fact that a point P(x,y) is equidistant from A(-1,2) and from B(3,4), we can use the distance formula. The distance between a point (x1, y1) and (x2, y2) is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's apply this formula to our situation. The distance from P(x,y) to A(-1,2) is equal to the distance from P(x,y) to B(3,4):
sqrt((x - (-1))^2 + (y - 2)^2) = sqrt((x - 3)^2 + (y - 4)^2)
Now, we can simplify this equation step by step:
(x + 1)^2 + (y - 2)^2 = (x - 3)^2 + (y - 4)^2
Expanding both sides:
x^2 + 2x + 1 + y^2 - 4y + 4 = x^2 - 6x + 9 + y^2 - 8y + 16
Simplifying:
2x + 2y - 7 = -6x - 8y + 25
Further simplifying:
8x + 10y = 32
Therefore, the equation that expresses the fact that a point P(x,y) is equidistant from A and from B is 8x + 10y = 32.
b) Geometrically, the set of points described by the equation 8x + 10y = 32 is a straight line. Since we have a linear equation with both x and y variables, it represents a line in the Cartesian coordinate plane.
If we plot this line on the coordinate plane, every point on the line will be equidistant from A and B. This means that the line represents all the points that have the same distance from A and B.