Let F1 = (0, 2), F2 = (0, −2), and P = (x, y). Use the distance formula to convert the
equation P F1 +P F2 = 6 into Cartesian form. Simplify your answer until it reaches the form
hx2 + ky2 = m
To convert the equation, we start by applying the distance formula:
PF1 = sqrt((x-0)^2 + (y-2)^2) = sqrt(x^2 + (y-2)^2)
PF2 = sqrt((x-0)^2 + (y+2)^2) = sqrt(x^2 + (y+2)^2)
Now we substitute these values back into the equation:
PF1 + PF2 = 6
sqrt(x^2 + (y-2)^2) + sqrt(x^2 + (y+2)^2) = 6
To eliminate the square root, we can square both sides of the equation:
(x^2 + (y-2)^2) + 2sqrt(x^2 + (y-2)^2)sqrt(x^2 + (y+2)^2) + (x^2 + (y+2)^2) = 36
Now we can simplify the equation:
2sqrt(x^2 + (y-2)^2)sqrt(x^2 + (y+2)^2) = 36 - 2(x^2 + (y-2)^2 + (y+2)^2)
2sqrt(x^2 + (y-2)^2)sqrt(x^2 + (y+2)^2) = 36 - 2(x^2 + y^2 - 4y + 4 + y^2 + 4y + 4)
2sqrt(x^2 + (y-2)^2)sqrt(x^2 + (y+2)^2) = 36 - 2(2x^2 + 2y^2 + 8)
Simplifying further:
2sqrt(x^2 + (y-2)^2)sqrt(x^2 + (y+2)^2) = 36 - 4x^2 - 4y^2 - 16
sqrt(x^2 + (y-2)^2)sqrt(x^2 + (y+2)^2) = 18 - 2x^2 - 2y^2
Now, we square both sides again to eliminate the square root:
(x^2 + (y-2)^2)(x^2 + (y+2)^2) = (18 - 2x^2 - 2y^2)^2
(x^2 + y^2 - 4y + 4)(x^2 + y^2 + 4y + 4) = (18 - 2x^2 - 2y^2)^2
Expanding and simplifying:
x^4 + 2x^2y^2 - 8x^2y + 8x^2 + y^4 + 8y^3 + 16y^2 + 16y + 4 = 324 - 72x^2 + 72y^2 + 4x^4 - 72xy^2 + 4y^4
3x^4 + 2x^2y^2 + 8x^2 - 72x^2 + 72y^2 + 8y^3 + 16y^2 + 16y + 4 = 324 - 72xy^2 + 4x^4 - 68y^2 + 4y^4
7x^4 + 2x^2y^2 - 72x^2 + 72y^2 + 8y^3 + 16y^2 + 16y - 72xy^2 = 320
Finally, we rewrite the equation in the form hx^2 + ky^2 = m:
7x^4 - 72x^2 + (2y^2 - 72y)x^2 + (72y^2 + 8y^3 + 16y^2 + 16y - 320) = 0
So, the Cartesian form is 7x^4 - 72x^2 + (2y^2 - 72y)x^2 + (72y^2 + 8y^3 + 16y^2 + 16y - 320) = 0