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Circle O is centered at the point of origin with point P=(3,4) lying on it. The red line l : 3x + 4y - 7 = 0 intersects the circle at points A and B, as shown.  What is the area of quadrilateral AOBP? Dec 28, 2020

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The radius of the circle  =   5

The equation of the  circle is   x^2 + y^2   = 25    (1)

Re-write  the equation of the line  as    y =(-3/4)x + 7/4    (2)

Sub (2) into (1)

x^2  +  ( 7/4  -(3/4)x)^2    =   25

x^2   + (1/16)(3x - 7)^2 =   25

16x^2  +  9x^2 - 42x + 49   =  400

25x^2  - 42x  -351   = 0

Solving this for  x  gives   x  = -3    and x  = 117/25

And we  only needt to find one associated value  for  y  because AP  = BP

So  A   = (-3  , (-3/4)(-3) + 7/4)   =  (-3, 4)

And AP  =   sqrt  [ ( -3 - 3)^2  +  (4 - 4)^2  ]  =    sqrt    =  6

And OP   = 5   and OA  = 5

So semi-perimeter of  OAP  =  [ 5 + 5 + 6] / 2    = 8

So....using symmetry  [AOBP ]  =

2sqrt   [ 8 * ( 8 - 5)^2  *  (8 - 6)  ]   =

2 sqrt  [ 8 * 9 * 2 ]   =

2sqrt [ 144 ] =

2 * 12   =

24   Dec 28, 2020