How do you write an equation of an ellipse in standard form given eccentricity of 0.5 if the vertices for the ellipse occur at the points ( 0 , 7 ) and ( 0, -9 )?

1 Answer
P dilip_k
Oct 19, 2016

Given that the vertices of an ellipse are (0,7) and (0,-9).
We can say that the center of the ellipse is #(0","(7-9)/2)=(0,-1)# and its major axis is on the y-axis. The length of major axis #2a=7-(-9)=16#
and semi major axis #a=8#

Now it is given that its eccentricity #(e)=0.5#

Again we know

#e^2=(a^2-b^2)/a^2#,where b is the length of semi minor axis.

So #(0.5)^2=(8^2-b^2)/8^2#

#=>1/4=1-b^2/64#

#=>b^2=64xx(1-1/4)=48#

#->b=4sqrt3#

So standard equation of the ellipse having center (0,-1) and semi major axis a=8 (along y-axis) and semi minor axis #b= 4sqrt3# (parallel to xaxis )is given by

#(x-0)^2/(4sqrt3)^2+(y-(-1))^2/8^2=1#

#=>x^2/(4sqrt3)^2+(y+1)^2/8^2=1#

#=>x^2/48+(y+1)^2/64=1#

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