Question

A circle passing through the vertex C of a rectangle ABCD and touching its sides AB and AD at M and N, respectively. If the distance from C to the line segment MN is equal to 5 units, then find the area of the reactangle ABCD.

Answer 1

Let us take AB and AD as coordinate axes.

If r is the radius of circle , then equation of circle touching both the axes is
`(x-r)^(2)+(y-r)^(2)=r^(2)`
or `x^(2)+y^(2)-2rx-2ry+r^(2)=0`
Let point C be (p,q)
Equation of line MN is `x+y-r=0`
Distance of MN from C is 5 units.
`:. (|p+q-r|)/(sqrt(2))=5`
`implies (p+q-r)^(2)=50`
Since (p,q) lies on the circle , we have
`p^(2)+q^(2)-2rp-2rq+r^(2)=0`
`implies (p+q-r)^(2)-2pq=0`
`implies 50-2pq=0`
`implies pq=25`
Therefore, area of rectangle `25sq.` units.