An ellipse slides between two perpendicular straight lines. Then identify the locus of its center.

Options

A. a circle
B. an ellipse
C. a parabola
D. a pair of straight lines

Answer

A

Explanation

Let 2a and 2 b be the length of the major axes of the ellipse which are parallel to the coordinate axes Let C(h,k) be the centre of the ellipse in one of its positions then its equation is
`((x-h)^(2))/(a^(2))+((y-k)^(2))/(b^(2))=1`
suppose this ellipse sidles between two perpendicular lines which intersect at `(alpha , beta )` this this means that `( alpha , beta)` is the point of intersection of perpendicular tangents to the ellipse , thus `(alpha , beta ) ` lies on the director circle of (i) whose equation is
`(x-h)^(2)+(y-K)^(2)=a^(2)+b^(2)`
`therefore (alpha -h)^(2)+( beta-k)^(2)=a^(2)+b^(2)`
thus the locus of (h,k) is
` ( alpha -x)^(2)+( beta - y)^(2)=a^(2)+b^(2)`
`or ( x-alpha)^(2)+(y-beta)^(2)=a^(2)+b^(2)`
which represents a circle with centre at `( alpha,beta)` and radius `sqrt(a^(2)+b^(2))`.