How do I solve this?
Describe the locus of points z satisfying:
|z+2|+|z-2|=5
z is a complex number.
To solve the equation |z+2|+|z-2|=5 and describe the locus of points satisfying it, we can break it down into cases based on the absolute values involved.
Case 1: z+2 and z-2 are both nonnegative or both nonpositive.
In this case, the equation simplifies to (z+2) + (z-2) = 5.
Simplifying further, we get 2z = 5, and thus z = 5/2.
So, when z+2 and z-2 have the same sign, the locus of points is a single point, namely z = 5/2.
Case 2: z+2 is nonnegative and z-2 is negative.
In this case, the equation simplifies to (z+2) - (z-2) = 5.
Simplifying further, we get 2z = 9, and thus z = 9/2.
So, when z+2 is nonnegative and z-2 is negative, the locus of points is a single point, namely z = 9/2.
Case 3: z+2 is negative and z-2 is nonnegative.
In this case, the equation simplifies to -(z+2) + (z-2) = 5.
Simplifying further, we get -2z = 5, and thus z = -5/2.
So, when z+2 is negative and z-2 is nonnegative, the locus of points is a single point, namely z = -5/2.
Case 4: z+2 and z-2 are both negative.
In this case, the equation simplifies to -(z+2) - (z-2) = 5.
Simplifying further, we get -2z = 9, and thus z = -9/2.
So, when z+2 and z-2 have the same sign, the locus of points is a single point, namely z = -9/2.
To summarize, the locus of points z satisfying the equation |z+2|+|z-2|=5 is a combination of four point solutions: z = 5/2, z = 9/2, z = -5/2, and z = -9/2.