Ellipse
Geometry
Level
pending
If S and T are the focii of an ellipse of major axis length 10 units and P is any point on the ellipse such that the perimeter of triangle PST is 15 units, then the eccentricity of the ellipse is
0.75
0.67
1
0.5
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1 solution
For an ellipse in standard position, and with its major axis along the x-direction, its equation is given by
a 2 x 2 + b 2 y 2 = 1 with a > b
The two foci are given by S = ( a e , 0 ) and T = ( − a e , 0 ) , where e is the eccentricity of the ellipse. In addition we have the following relation,
S P + T P = 2 a
Therefore, the perimeter of the triangle is given by
S P + T P + S T = 2 a + 2 a e = 2 a ( 1 + e ) = 1 0 ( 1 + e ) = 1 5
From which e = 0 . 5 .