Finding the equation of the locus of a moving point

Question

The sum of the distance of a moving point, P(x,y) from the points (c,0) and (-c,0) is always 2a units. How can I find the equation of the locus of the moving point?

Answer 1

This is the definition of the ellipse with foci (c,0) and (-c,0), with center (0,0), and with horizontal major axis of length 2a.

Equation: (x-0)² / a² + (y - 0)² / b² = 1, where b² = a² - c². Simplify to get x² / a² + y² / b² = 1.

You can also use the distance formula to derive the equation above:

√[(x-c)² + (y-0)²] + √[(x+c)² + (y-0)²] = 2a

Answer 2

the problem specifies "moving point"

x(t)=acos(kt)

y(t)=bsin(kt)

t in seconds

k in radian/second

a is given

b=√(a²-c²)

where c is given

produces a moving point ellipse locus