The radius of the circle passing through the two foci and an end point of the minor axis of the ellipse  x 2/16 + y 2/25 =1 is ?

Asked by Malavika Umesh | 1st May, 2015, 01:52: PM

Expert Answer:

For the ellipse x squared over 16 plus y squared over 25 equals 1, the major axis is along Y-axis and the minor axis is along X-axis. (since the denominator of x2 term is less)
Here a squared equals 25 comma space b squared equals 16
c squared equals a squared minus b squared equals 25 minus 16 equals 9
The coordinates of foci are open parentheses 0 comma space plus-or-minus c close parentheses equals open parentheses 0 comma space plus-or-minus 3 close parentheses
The end points of minor axis are open parentheses plus-or-minus b comma space 0 close parentheses equals open parentheses plus-or-minus 4 comma space 0 close parentheses
So, we have to form a circle passing through open parentheses 0 comma space plus-or-minus 3 close parentheses and one end point of minor axis open parentheses 4 comma space 0 close parentheses.
Since, it's an isosceles triangle, the center lies on the perpendicular bisector of open parentheses 0 comma space plus-or-minus 3 close parentheses, that is X-axis.
Let the coordinates of the center be open parentheses p comma 0 close parentheses.
Equating the square of radii of the circle we get
p squared plus 3 squared equals open parentheses 4 minus p close parentheses squared rightwards double arrow p squared plus 3 squared equals 16 plus p squared minus 8 p rightwards double arrow 8 p equals 7 rightwards double arrow p equals 7 over 8
Hence, radius=4 minus 7 over 8 equals 25 over 8

Answered by satyajit samal | 3rd May, 2015, 10:43: AM