ABCD is a rectangle in which AB = 2/3 units and AD=1unit. A circle passes through A and B , and touches CD at its mid-point. Find the area of the shaded region.
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give the shaded region and then I will give
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Let us take AB and AD as coordinate axes If r be the radius of circle, then its centre is P(r,r) Equation of circle is (x−r)2+(y−r)2=r2
⇒x2+y2−2rx−2ry+r2=0
Let the coordinates of C≡(p,q)
Equation of MN is x+y=r Its distance from C is 5 units ⇒2∣p+q−r∣=5
⇒(p+q−r)2=50
since (p,q) lie on the circle,⇒
p2+q2−2rp−2rq+r2=0
(p+q−r)2−2pq=0
⇒50−2pq=0⇒pq=25
Area of rectangle =25 sq. units
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