A circle is tangent to the x and y axes in the first quadrant at the points P and Q respectively. BC and AD are parallel tangents to the circle with slope −1. If the points A and B are on the y- axis while C and D are on the x-axis and the area of the quadrilateral ABCD is 900 2 sq. units, then the radius of the circle is A. 10 B. 12 C. 15 D. 20

Question

A circle is tangent to the x and y axes in the first quadrant at the points P and Q respectively. BC and AD are parallel tangents to the circle with slope −1. If the points A and B are on the y- axis while C and D are on the x-axis and the area of the quadrilateral ABCD is   900      2   sq. units, then the radius of the circle is A. 10 B. 12 C. 15 D. 20

Sydnee Glover · Accepted Answer

Radius   =      α    ,      α    ∈  R Centre   =  (      α    ,      α    )  M  N  ⊥  B  C  ,  M  N  ⊥  A  D MN equation is y=x. The distance form of the MN equation is            x      −              α                    cos      ⁡              45        ∘              =            y      −              α                    sin      ⁡              45        ∘              =      α    ⇒  x  =      α    +            α              2        =  y∴ Coordinates of M is   (      α    +            α              2        ,      α    +            α              2        ) Similarly, coordinates of N is   (      α    −            α              2        ,      α    −            α              2        )x intercept of AD is   2      α    +      2        α  x intercept of BC is   2      α    −      2        α   Area of trapezium   A  B  C  D  =      1    2    (  A  D  +  B  C  )  M  N  ⇒  900      2    =      1    2    ×      2    (  4      α    )  ×  2      α      ⇒      α    =  15

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