This is the equation of an ellipse of the form (x-h)^2/a^2+(y-k)^2/b^2=1(x−h)2a2+(y−k)2b2=1, whose center is (h,k)(h,k), major axis is 2a2a and minor axis is 2b2b. Vertices on major axis are (h+-a,k)(h±a,k) and along minor axis are (h,k+-b)(h,k±b). Eccentricity is given by e=sqrt(1-b^2/a^2)e=√1−b2a2 and focii are (h+-ae,k)(h±ae,k).
As x^2/169+y^2/25=1x2169+y225=1 can be written as
(x-0)^2/13^2+(y-0)^2/5^2=1(x−0)2132+(y−0)252=1
Hence, this is an equation of an ellipse, whose center is (0,0)(0,0), major axis is 13xx2=2613×2=26 and minor axis is 5xx2=105×2=10.
Vertices are (-13,0),(13,0),(5,0)(−13,0),(13,0),(5,0) and (-5,0)(−5,0).
Eccentricity is e=sqrt(1-5^2/13^2)=sqrt(12^2/13^2)=12/13e=√1−52132=√122132=1213
and fociie are (+-(13xx12/13),0)(±(13×1213),0) i.e. (-12,0)(−12,0) and (12,0)(12,0).
We can mark the four vertices, if so desired a few more points by using the equation x^2/169+y^2/25=1x2169+y225=1. For example, if x=+-3x=±3. y=+-5sqrt(1-9/169)=+-20sqrt(10/169)=+-4.865y=±5√1−9169=±20√10169=±4.865 gives four more points (3,4.865),(-3,4.865), (3,-4.865)(3,4.865),(−3,4.865),(3,−4.865) and (-3,-4.865)(−3,−4.865), joining which along with vertices will give us the desired ellipse.
The ellipse appears as shown below:
graph{(x^2/169+y^2/25-1)((x+13)^2+y^2-0.04)((x-13)^2+y^2-0.04)(x^2+(y+5)^2-0.04)(x^2+(y-5)^2-0.04)((x+12)^2+y^2-0.04)((x-12)^2+y^2-0.04)=0 [-20, 20, -10, 10]}
