This is about ellipses:
If a problem gives you two foci points and the length of a spring, how do you find the equation and the vertices of the ellipse?
Also, if you have only the equation of an ellipse and the foci points, how do you find the length of the string?
spring? or string?
Pretty easy. Put the string at each foci up to the top of the semimajor axis.
LenghString^2=)2*focidistance )^2+semimajoraxis^2
from that, find the semimajor azis.
Next, put the string horizontal to the major axis.
Length string=2semimaforaxis+majoraxis
solve for major axis length.
To find the equation and vertices of an ellipse given two foci points and the length of a spring, you can follow these steps:
1. Understand the definition of an ellipse: An ellipse is a closed curve in which the sum of the distances from any point on the curve to two fixed points (called the foci) is constant.
2. Identify the given information:
- Determine the coordinates of the two foci points. Let's use (F1, F2) and (F3, F4) to represent these coordinates.
- Find the length of the spring. Let's call this value 2a.
3. Use the distance formula to find the distance between the foci points: The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula √((x₂ - x₁)² + (y₂ - y₁)²). Apply this formula to find the distance between the two foci points: √((F3 - F1)² + (F4 - F2)²). Let's call this value 2c.
4. Identify the center of the ellipse: The center of the ellipse is the midpoint between the two foci points. Find the x-coordinate of the center by calculating (F1 + F3)/2 and the y-coordinate of the center by calculating (F2 + F4)/2. Let's denote the center coordinates as (h, k).
5. Find the value of b: To find the value of b, which represents the semi-minor axis of the ellipse, you can use the Pythagorean theorem. Use the equation b² = a² - c², where a = c + b. Rearrange this equation to solve for b: b = √(a² - c²).
6. Determine the length of the major axis and the minor axis: The length of the major axis is equal to 2a, which is given in the problem as the length of the spring. The length of the minor axis is equal to 2b.
7. Write the equation of the ellipse: The standard form for the equation of an ellipse centered at (h, k) is ((x - h)² / a²) + ((y - k)² / b²) = 1. Substitute the values of a, b, h, and k into the equation to get the final form of the equation.
8. Find the vertices of the ellipse: The vertices of an ellipse are the points where the ellipse intersects the major axis. Since the major axis is parallel to the x-axis, the vertices will have the coordinates (h ± a, k).
Following these steps will allow you to find the equation and vertices of the ellipse given the two foci points and the length of the spring.