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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.