Lesson Derive an equation and show that the locus of points is an ellipse

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Derive an equation and show that the locus of points is an ellipse

Problem 1

What is the equation of the locus of the point which moves so that its distance from the point (2,0)
is 2/3 its distance from the line y = 5 ?

Solution

Let (x,y) be the point of the locus.


Then the distance to point (2,0) is  d = ,

while the distance to the line y=5 is  |y-5|.


Equate the equal distances


     = .   


It is the basic equation which translates the input into mathematical form.


Next, square both sides of this equation and reduce it to the standard conic section equation.


     =     

     = 

     = 

     +   = 64

     +  = 64 + 36 + 80

     +  = 180.

     +  = 1


It is the standard form equation of the ellipse centered at the point  (2,-4)  and

the major semi-axis   = 6 (vertical)  and the minor semi-axis   =   (horizontal).