Question

Find the equation in standard form of the ellipse whose foci are F1(-8,0) and F2(8,0), such that for any point on it, the sum of its distance from the foci is 20.

a) (x + 8)²/64 + y²/36 = 1
b) x²/64 + y²/36 = 1
c) (x - 8)²/64 + y²/36 = 1
d) x²/36 + y²/64 = 1

Answer 1

The equation of the ellipse in standard form with the given foci and sum of distances to any point being 20 is x²/64 + y²/36 = 1.

Step-by-step explanation:

The student asks for the equation of an ellipse in standard form, given that the foci are F1(-8,0) and F2(8,0), with the sum of the distances to any point on the ellipse being 20. The standard form of the ellipse equation is x²/a² + y²/b² = 1, where 2a is the sum of the distances from any point on the ellipse to the foci (which here is 20), so a is 10 and a² is 100. The distance between the foci is 2c, where each focus is 'c' units from the center, so in this case 2c is 16 and c² is 64. To find b², we use the relationship c² = a² - b², which yields b² is 36. Therefore, the equation of the ellipse is x²/64 + y²/36 = 1.