Posted by **Anal-G** on Friday, January 14, 2011 at 8:11am.

Find the equation of a locus of a point which moves so that the sum of its distance from (2,0)and (-2,0) is 8.

it is about equation of a locus..

please help!!

- Analytic Geometry -
**Reiny**, Friday, January 14, 2011 at 9:12am
Your description defines an ellipse.

does the question expect you to find that equation by using that definition?

if so, then let such a point on that locus be P(x,y)

√[(x-2)^2 + y^2] + √[(x+2)^2 + y^2] = 8

√[(x-2)^2 + y^2] = 8 - √[(x+2)^2 + y^2]

square both sides and expand

x^2 - 4x + 4 + y^2 = 64 - 16√[(x+2)^2 + y^2] + x^2 + 4x + 4 + y^2

16√[(x+2)^2 + y^2] = 64 + 8x

2√[(x+2)^2 + y^2] = 8 + x

squaring again and simplifying I get

3x^2 + 4y^2 = 48

divide each term by 48 to get it into standard form

x^2/16 + y^2/12 = 1

or, the easy way

from the description 2a = 8 , a = 4

(2,0) and (-2,0) must be the focal points so the midpoint or (0,0) must be the centre and c = 2

since the focal points lie on the x-axis

b^2 + c^2 = a^2

b^2 + 4 = 16

b^2 = 12

standard form with centre (0,0) is

x^2/a^2 + y^2/b^2 = 1

so

x^2/16 +y^2/12 = 1

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