SOLUTION: Find an equation for the collection of points for which the distance to (6, 0) is twice the distance to the line x = -6. Must be in the form {{{ ((x-h)^2)/(p^2)-((y-k)^2)/(q^2) =

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Find an equation for the collection of points for which the distance to (6, 0) is twice the distance to the line x = -6. Must be in the form {{{ ((x-h)^2)/(p^2)-((y-k)^2)/(q^2) =      Log On


   

Question 1188404: Find an equation for the collection of points for which the distance to (6, 0) is twice the distance to the line x = -6. Must be in the form
+%28%28x-h%29%5E2%29%2F%28p%5E2%29-%28%28y-k%29%5E2%29%2F%28q%5E2%29+=+1+
Answer by ikleyn(52800) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find an equation for the collection of points for which the distance to (6, 0)
is twice the distance to the line x = -6.
~~~~~~~~~~~~~~~ 

Let (x,y) be the point in the coordinate plane from this collection of points.


The distance from (x,y) to the point (6,0) is

    d = sqrt%28%28x-6%29%5E2%2B%28y-0%29%5E2%29 = sqrt%28%28x-6%29%5E2+%2B+y%5E2%29,


The distance from (x,y) to the line  x= -6  is  |x+6|  (notice the absolute value sign (!) )


From the problem, we have this equation

    sqrt%28%28x-6%29%5E2+%2B+y%5E2%29 = 2*|x+6|.


Square both sides

    (x-6)^2 + y^2 = 4(x+6)^2

and simplify

    x^2 - 12x + 36 + y^2 = 4x^2 + 48x + 144

    3x^2 + 60x - y^2 = -108

    3(x^2 + 20x + 100) - y^2 = -108 + 300

    3(x+10)^2 - y^2 = 192

    %28x%2B10%29%5E2%2F8%5E2 - y%5E2%2F192 = 1


It is your answer (= final equation).  p = 8;  q = sqrt%28192%29 = sqrt%2864%2A3%29 = 8%2Asqrt%283%29.

Solved.