Question

A circle touches the sides AB and AD of the rectangle ABCD at P and Q respectively and passes through the vertex C. If the distance of C from chord PQ is 5 units and the area of the rectangle is $\lambda$, then find the value of $\frac{\lambda }{5}$

• A. 2
• B. 5
• C. 10
• D. 25

Answer 1

The correct option is B 5

Let AB and AD be along the x-axis and y-axis respectively. Let $AB=a$ and $AD=b$ let (h,h) be the center of the circle. Then equation of PQ is x+y=h.
its distance from (a,b)$=\frac{a+b-h}{\sqrt{2}}=5$
also $h=\sqrt{{(a-h)}^2+{(b-h)}^2}$
i.e. $h^2=a^2+b^2-2ah-2bh+2h^2$.......(i)
i.e. $|a^2+b^2-h|=5\sqrt{2}$
$a^2+b^2+h^2+2ab-2ah-2bh=50$
$\therefore 2ab=50$
$\therefore ab=25$