Proving the Equation ( z - 3i 8 ) Represents an Ellipse

In this article, we will analyze and prove that the equation ( z - 3i 8 ) represents an ellipse in the complex plane. We will follow a step-by-step approach to understand the geometric interpretation and verify the conditions that define an ellipse.

Step 1: Understanding the Components

Let us denote the complex number ( z ) in the form ( z x yi ), where ( x ) and ( y ) are real numbers. The terms ( z - 3i ) and ( z 3i ) can be rewritten as follows:

1. ( z - 3i x y - 3i sqrt{x^2 (y - 3)^2} )

2. ( z 3i x y 3i sqrt{x^2 (y 3)^2} )

Step 2: Interpreting the Equation

The equation ( |z - 3i| |z 3i| 8 ) can be interpreted geometrically. The expression ( |z - 3i| ) represents the distance from the point ( (x, y) ) to the point ( (0, -3) ), which corresponds to ( z -3i ), and ( |z 3i| ) represents the distance from the point ( (x, y) ) to the point ( (0, 3) ), which corresponds to ( z 3i ).

Step 3: Definition of an Ellipse

An elliptical curve in the complex plane is defined as the set of points ( P ) such that the sum of the distances from the two fixed points (foci) is constant. In this case, the foci are at ( (0, -3) ) and ( (0, 3) ), and the constant distance is 8.

Step 4: Verifying the Conditions

Foci: The foci of the ellipse are at ( F_1(0, -3) ) and ( F_2(0, 3) ).

Constant Distance: The sum of the distances from any point ( (x, y) ) on the ellipse to the two foci is 8.

To verify this, we can use the distance formula. The distance between the two foci is given by:

( d sqrt{(0 - 0)^2 (3 - (-3))^2} sqrt{6^2} 6 )

For an ellipse, the sum of the distances from any point on the ellipse to the foci must be greater than the distance between the foci. Here, ( 8 > 6 ), which confirms that it satisfies the condition for being an ellipse.

Step 5: Checking the Length of the Major Axis

The length of the major axis of the ellipse is determined by the constant distance, which is 8. The major axis is aligned along the y-axis due to the symmetry of the foci.

The length of the major axis is 8, and the distance between the foci is 6. Therefore, the semi-major axis ( a ) is 4, and the semi-minor axis ( b ) can be calculated using the relationship ( b^2 a^2 - c^2 ), where ( c ) is the distance from the center to each focus.

Here, ( a 4 ) and ( c 3 ). So,

( b^2 4^2 - 3^2 16 - 9 7 )

( b sqrt{7} )

Conclusion

Since the equation ( |z - 3i| |z 3i| 8 ) describes the set of points for which the sum of the distances to the two foci ( (0, -3) ) and ( (0, 3) ) is a constant 8, we conclude that this equation represents an ellipse.

Final Verdict: The given equation ( z - 3i 8 ) is indeed an ellipse in the complex plane, and its geometric interpretation confirms its validity.