if you have the ellipse (x^2/a^2)+(y^2/b^2)=1 and a>b what would the sum of the distances from a point (x,y) on the ellipse to each of the foci be.
going along with the above problem how would you determine two formulae that calculates the distance from a point on the ellipse to each of the foci. please help
by definition , the sum of the distances from a point P(x,y) to each of the focal points is
2a
see http://mathworld.wolfram.com/Ellipse.html
if you want the individual distances, then first find the foci which would be (c,0) and (-c,0) by using
a^2 = b^2 + c^2, for a>b and then using the distance between two points formula
distance = √((x-c)^2 + y^2)) and
distance = √((x+c)^2 + y^2))
e.g.
for x^2/25 + y^2/9 = 1
a=5, b=3, the by 5^2 = 3^2 + c^2
c = +/- 4
so the foci are (4,0) and (-4,0)
a point P could be (2,(√(189)/5))
(I got that by subbing x=2 into my equation and solving for y)
then
distance#1 = √(2-4)^2 + (√(189)/5 - 0)^2) = √(289/25) = 3.4
distance #2 = √(2+4)^2 + (√(189)/5 - 0)^2) = √(1089/25) = 6.6
notice 3.4 + 6.6 = 10 or 2a
To find the sum of the distances from a point (x, y) on the ellipse to each of the foci, we can use the formula for the distance between two points in a coordinate system:
d = √[(x₁ - x₂)² + (y₁ - y₂)²]
Here, we need to calculate the distances from the point (x, y) on the ellipse to each of the foci. Since the ellipse is defined by the equation (x²/a²) + (y²/b²) = 1, we know that the center of the ellipse is at the origin (0, 0).
The distance between the point (x, y) on the ellipse and each focus can be found using the relationship between the foci and the major/minor axes of the ellipse. For an ellipse, the distance between the center (0, 0) and each focus is given by c, where c is calculated using the following formula:
c = √(a² - b²)
Since a > b in your case, we can calculate the distances of each focus from the point (x, y) on the ellipse as follows:
Distance to Focus 1: d₁ = √[(x - c)² + y²]
Distance to Focus 2: d₂ = √[(x + c)² + y²]
Finally, to find the sum of these two distances, we can add them together:
Sum of Distances = d₁ + d₂
Remember to substitute the value of c using c = √(a² - b²), based on the given ellipse equation.
Make sure to substitute the values of a, b, x, and y from the given problem to find the exact sum of distances.