Question

How do you find the equation of the following conic section and identify it given: All points such that the sum of the distance to the points (3,1) and (-1,1) equals 6?

Answer 1

Equation is `5x^2+9y^2-10x-18y-31=0` and is of an ellipse.

Explanation:

Let the point on the locus be `(x,y)`, sum of whose distances from `(3,1)` and `(-1,1)` is `6`, hence

`sqrt((x-3)^2+(y-1)^2)+sqrt((x+1)^2+(y-1)^2)=6` or

`sqrt((x-3)^2+(y-1)^2)=6-sqrt((x+1)^2+(y-1)^2)`

Squaring each side, we get

`(x-3)^2+(y-1)^2=36+(x+1)^2+(y-1)^2-12sqrt((x+1)^2+(y-1)^2)` or

`x^2-6x+9+y^2-2y+1=36+x^2+2x+1+y^2-2y+1-12sqrt((x+1)^2+(y-1)^2)` or

`-6x+9=36+2x+1-12sqrt((x+1)^2+(y-1)^2)` or

`12sqrt((x+1)^2+(y-1)^2)=36+2x+1+6x-9=8x+28` or

`3sqrt((x+1)^2+(y-1)^2)=2x+7` and squaring again

`9((x+1)^2+(y-1)^2)=4x^2+28x+49` or

`9(x^2+2x+1+y^2-2y+1)=4x^2+28x+49` or

`5x^2+9y^2-10x-18y-31=0`

As the coefficient of `x^2` and `y^2` are positive but different, it is an ellipse.

graph{5x^2+9y^2-10x-18y-31=0 [-10, 6, -5, 5]}