Given the ellipse with foci at (5,3) and (5,-5) and a major axis length of 12, determine the Center and the denominators of the equation:
Center - (h,k)
(x )^2 (y )^2 = 1
Determine the denominators of the above equation
Given two fixed points (foci) `F_1(5,3) and F_2(5,-5)` .Observe ordinates of fixed points are different ,it means major axis lies parallel to y -axis .
Equation of the major axis is
Center of the ellipse is mid point of `F_1 and F_2` i.e.`((5+5)/2,(3-5)/2)=(5,-1)`
the length of major axis =12
Let (x,y) be the point on the ellipse . Thus By def. of the ellipse ,we have ,
squaring both side and simplify ,we get
`` squaring again and simplify, we have
The ellipse has its foci at (5, 3) and (5,-5).
The foci have different y-coordinates indicating that the foci lie in the y-axis. Therefore, it is a vertical ellipse, having its major axis parallel to the y-axis. The standard form of the equation of the ellipse is:
with its center at (h, k), and a ≥ b > 0.
For a vertical ellipse, coordinates of the two foci are (h, k+c) and (h, k-c), where c is the focal distance given by `c=sqrt(a^2-b^2)` . Hence the y-coordinate of the center will be the mid-point of the y-coordinates of the two foci.
Center of the ellipse is at (5, -1).
Comparing the coordinates of the foci, we get the equations:
k+c=3 and k-c=-5, where k=-1
Upon solving, c=4.
Again, the length of the major axis, 2a=12
Putting this value of a in the expression for c, we get
Therefore, the required equation of the ellipse is: