Answer :
To write the standard form of the equation of an ellipse given the foci and co-vertices, follow these steps:
1. Identify the distance from the center to the foci (c): The foci are given as (±5,0). This means that the distance from the center of the ellipse to each focus is 5 units on the x-axis. So c = 5.
2. Identify the distance from the center to the co-vertices (b): The co-vertices are given as (0, ±5). This means that the distance from the center of the ellipse to each co-vertex is 5 units on the y-axis. So b = 5.
3. Find the distance from the center to the vertices (a): Since the foci and co-vertices are equidistant from the center, and they lie along the axes, we conclude this ellipse has a circular shape where both axes have the same length. Therefore, the distance from the center to the vertices (a) is equal to the distance from the center to the co-vertices (b), thus a = b = 5.
4. Write the standard form of an ellipse equation:
When a = b, the ellipse is actually a circle, and its standard form equation is,
[tex]\[x^2 + y^2 = r^2\][/tex]
where r is the radius of the circle (distance from center to a vertex).
Since a and b are both 5, the radius r is also 5, and the equation simplifies to:
[tex]\[x^2 + y^2 = 5^2\][/tex]
[tex]\[x^2 + y^2 = 25\][/tex]
This is the standard form for the equation of a circle, which is a special case of an ellipse where the major and minor axes are equal.
1. Identify the distance from the center to the foci (c): The foci are given as (±5,0). This means that the distance from the center of the ellipse to each focus is 5 units on the x-axis. So c = 5.
2. Identify the distance from the center to the co-vertices (b): The co-vertices are given as (0, ±5). This means that the distance from the center of the ellipse to each co-vertex is 5 units on the y-axis. So b = 5.
3. Find the distance from the center to the vertices (a): Since the foci and co-vertices are equidistant from the center, and they lie along the axes, we conclude this ellipse has a circular shape where both axes have the same length. Therefore, the distance from the center to the vertices (a) is equal to the distance from the center to the co-vertices (b), thus a = b = 5.
4. Write the standard form of an ellipse equation:
When a = b, the ellipse is actually a circle, and its standard form equation is,
[tex]\[x^2 + y^2 = r^2\][/tex]
where r is the radius of the circle (distance from center to a vertex).
Since a and b are both 5, the radius r is also 5, and the equation simplifies to:
[tex]\[x^2 + y^2 = 5^2\][/tex]
[tex]\[x^2 + y^2 = 25\][/tex]
This is the standard form for the equation of a circle, which is a special case of an ellipse where the major and minor axes are equal.
