Center at (2,-5), conjugate axis parallel to the y-axis, slopes at assymptotes numerically one-sixteenth times the length of the latus rectum, and distance between foci is 2 sqr t145.
To find the equation of an ellipse given the information provided, we can follow these steps:
Step 1: Identify the center coordinates
Given that the center is at (2, -5), we have the values for h and k - the x-coordinate and y-coordinate of the center.
Step 2: Determine the lengths of the major and minor axes
Since the conjugate axis is parallel to the y-axis, its length is the diameter of the ellipse. We need the length of the latus rectum, which is the chord of the ellipse passing through its foci and perpendicular to the major axis.
The length of the latus rectum is given by the equation: l = 2b^2/a
Given that the slope at the asymptotes is numerically one-sixteenth times the length of the latus rectum, we have:
slope = l/16
b = (a * slope)/2
where a is the length of the semi-major axis.
Step 3: Determine the value of a
We are given the distance between the foci is 2√t145. The formula for the distance between the foci is given by the equation:
c^2 = a^2 - b^2
where c is the distance between the center and each focus.
Plugging in the values, we have:
(2√t145)^2 = a^2 - b^2
4t145 = a^2 - b^2
But b = (a * slope)/2, so we can substitute that in the equation:
4t145 = a^2 - (a * slope/2)^2
4t145 = a^2 - (a^2 * slope^2)/4
4t145 = (4a^2 - a^2 * slope^2)/4
16t145 = 4a^2 - a^2 * slope^2
16t145 = 3a^2 - a^2 * slope^2
16t145 = a^2 * (3 - slope^2)
a^2 = 16t145 / (3 - slope^2)
Since we know the value of slope, we can substitute that in and calculate the value of a.
Step 4: Determine the value of b
Now that we have the value of a, we can substitute it into the equation b = (a * slope)/2 to calculate the value of b.
Step 5: Write the equation of the ellipse
Now that we have the values of the center (h, k), the lengths of the major and minor axes (2a and 2b), we can write the equation of the ellipse in standard form:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Substituting the values of h, k, a, and b into the equation will give us the final equation of the ellipse.
By following these steps, you can find the equation of the ellipse.