write an equation in standard form for the ellipse with foci (7,0) and (-7,0) and y intercepts 6 and -6
center at origin by symmetry
x^2/a^2 + y^2/b^2 = 1
given that b = 6 center to focus = 7
but 7^2 = a^2-b^2
49 = a^2 - 36
a^2 = 85
a = sqrt 85
so
x^2/85 + y^2/36 = 1
To get the equation of an ellipse in standard form, we need to know the coordinates of the foci and the lengths of the major and minor axes.
Given that the foci are at (7, 0) and (-7, 0), we can determine that the length of the major axis (2a) is equal to the distance between the foci, which is 2a = 7 - (-7) = 14. Therefore, a = 7.
The y-intercepts give us the values of b and c, where b is half the length of the minor axis, and c is the distance from the center of the ellipse to each focus. In this case, we have b = 6 and -6, so we can conclud that b = 6.
To find c, we can use the equation c^2 = a^2 - b^2, which relates a, b, and c. Substituting the given values, we have c^2 = 7^2 - 6^2, resulting in c^2 = 49 - 36 = 13. Taking the square root of both sides, we find c = √13.
Now we have all the necessary information to determine the equation of the ellipse in standard form, which is:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
In this case, since the foci are located at (7, 0) and (-7, 0), the center of the ellipse is at the origin (0, 0), so h = 0 and k = 0.
Plugging in the values of a, b, h, and k in the equation, we get:
(x - 0)^2/7^2 + (y - 0)^2/6^2 = 1
Simplifying the equation further:
x^2/49 + y^2/36 = 1
Therefore, the equation in standard form for the given ellipse is x^2/49 + y^2/36 = 1.
To write the equation of an ellipse in standard form, we need to determine the center, major radius, and minor radius of the ellipse.
Step 1: Determine the center:
The center of the ellipse is the midpoint between the two foci. In this case, the foci are (7,0) and (-7,0), so the center is obtained by taking the average of the x-coordinates and the y-coordinates. Therefore, the center of the ellipse is (0, 0).
Step 2: Determine the major and minor axes:
The distance between the center and each focus is the half of the major axis length, represented by "a". In this case, the distance between (0,0) and (7,0) is 7. Therefore, the major axis length, 2a, is equal to 14, and a is equal to 7.
The distance between the center and each y-intercept is the half of the minor axis length, represented by "b". In this case, the distance between (0,0) and (0,6) is 6. Therefore, the minor axis length, 2b, is equal to 12, and b is equal to 6.
Step 3: Write the equation in standard form:
The standard form equation for an ellipse is:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
Where (h, k) represent the coordinates of the center.
Substituting the values we found, the equation becomes:
(x-0)^2/7^2 + (y-0)^2/6^2 = 1
Simplifying the equation further, we have:
x^2/49 + y^2/36 = 1
Therefore, the equation in standard form for the given ellipse is:
x^2/49 + y^2/36 = 1