Using the fact that |z 1 - z 2 | is thedistance between two points z 1 and z 2 ,

Question

Using the fact that |z1 - z2| is thedistance between two points z1 and z2, give ageometric argument that a) |z-4i| + |z + 4i| = 10 represents an ellipse whose foci are(0,+-4) b) |z-1| = |z + i| represents the line throught the originwhose slope is -1. Using the fact that |z1 - z2| is thedistance between two points z1 and z2, give ageometric argument that a) |z-4i| + |z + 4i| = 10 represents an ellipse whose foci are(0,+-4) b) |z-1| = |z + i| represents the line throught the originwhose slope is -1.

Explanation / Answer

(a) The set of all points such that the sum of the distancesto the points 4i and -4 is a constant(10) is the geometricdefinition of an ellipse with foci at the two fixed points. (b) The set of all points which are equidistant to two fixedpoints :1 and -i is the perpendicular bisector of the line segmentformed by the two points.The slope of the perpendicular is theopposite reciprocal of the slope of the line connecting the twopoints, the line connecting 1 and -i has a slope of one, theperpendicular bistor has a slope of -1.

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