Question

An ellipse hase semi-major of length 2 and semi-minor axis of length 1. It slides between the coordinates axes in the first quadrant while mantaining contact with both x-axis and y-axis. The locus of the centre of the ellipse is
A. `x^(2)+y^(2)=8`
B. `(x^(2))/(9)+(y^(2))/(4)=1`
C. `y^(2)=4x`
D. `x^(2)+y^(2)=5`

Answer 1

Let the centre of the ellipse be C(x,y)
Clearly, axeas are tangents to the ellipse to the ellipse which are perpendicular and intesect at origin O.
So, origin lies on the director circle
Thus, origin lies on the director circle.
OC is radius of he director circle .
So, `OC^(2)=a^(2)+b^(2)=22+12`
or `x^(2)+y^(2)=5`