Question

A circle is drawn with the two foci an ellipse `x^(2)/a^(2)+y^(2)/b^(2)=1` at the end of the diameter. What is the equation of the eircle ?
A. `x^(2)+y^(2)=a^(2)+b^(2)`
B. `x^(2)+y^(2)=a^(2)-b^(2)`
C. `x^(2)+y^(2)=2a^(2)+b^(2)`
D. `x^(2)+y^(2)=2a^(2)-b^(2)`

Answer 1

Correct Answer - B
Foci of an ellipse `x^(2)/a^(2)+y^(2)/b^(2)=1` are given as(ae, 0) and (-ae, 0).
Since, two foci are at the end of the diameter
`:." Equation of circle, is "`
`(x-ae)(x+ae)+(y-0)(y-0)=0`
`rArr" "x^(2)-a^(2)e^(2)+y^(2)=0`
`rArr" "x^(2)+y^(2)-a^(2)(1-b^(2)/a^(2))=0(becausee=sqrt(1-b^(2)/a^(2)))`
`rArr" "x^(2)+y^(2)-a^(2)+b^(2)=0`
`rArr" "x^(2)+y^(2)=a^(2)-b^(2)`