for the last one,

i hope you know for an ellipse .

where 2c=focal distance, 2b=minor axis length, 2a=major axis length.

So having origin as centre for the ellipse, the point (c,y) lies on the ellipse for some y which we will find.

therefore, placing (c,y) in standard ellipse eqn:

.

using , we get .

thus distance of point from (c,0) i.e. a focus, is which is the semi-latus rectum.

in case u dont know .

from defn. of ellipsesum of distances of a point from two focii of an ellipse is equal to length of major axis. So, for a standard ellipse having origin as centre, for the point (0,b),

dist of (0,b) from (-c,0)=dist of (0,b) from (c,0)=a. [from geometry]

the triangle formed by (0,0),(c,0),(0,b) is right angled. So from Pythagoras, .