Answer:

Step-by-step explanation:
We want to find the locus of a point such that the sum of the distance from any point P on the locus to (0, 2) and (0, -2) is 6. 
First, we will need the distance formula, given by:

Let the point on the locus be P(x, y). 
So, the distance from P to (0, 2) will be: 

And, the distance from P to (0, -2) will be: 

So sum of the two distances must be 6. Therefore: 

Now, by substitution:

Simplify. We can subtract the second term from the left: 

Square both sides: 

We can cancel the x² terms and continue squaring: 

We can cancel the y² and 4 from both sides. We can also subtract 4y from both sides. This leaves us with: 

We can divide both sides by -4: 

Adding 9 to both sides yields: 

And, we will square both sides one final time. 

Distribute: 

The 36y will cancel. So: 

Subtracting 4y² and 36 from both sides yields: 

And dividing both sides by 45 produces: 

Therefore, the equation for the locus of a point such that the sum of its distance to (0, 2) and (0, -2) is 6 is given by a vertical ellipse with a major axis length of 3 and a minor axis length of √5, centered on the origin.