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P and Q are two points on line x - y + 1=0 and are at distance 5 units from the origin. Find the area of the triangle OPQ

 

 Dec 31, 2020
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We just need to  find  the intersections  of  the circle with the equation   x^2 + y^2   =25   and the line 

x - y + 1   =  0 ⇒   y   =  x + 1

 

So

 

x^2  +  ( x + 1)^2   =25

 

x^2  + x^2  + 2x  +  1    = 25

 

2x^2  + 2x  - 24  =   0

 

x^2  + x  -  12  =   0       factor

 

(x + 4)(x - 3)   = 0

 

Set  each factor to 0  and solve for x

 

x = -4    and  x  =  3

 

When x  = -4   y  = -3

When x =  3  , y  = 4

 

 

The distance between these points is    sqrt  ( (3 - -4)^2  + (4  - - 3)^2 )  =  sqrt ( 2 * 49)  = 7sqrt (2)

 

The semi-perimeter    =     (10 + 7sqrt (2)) / 2)  =   5 + 3.5sqrt (2)

 

The area  =   sqrt [ ( 5 + 3.5sqrt (2) )  ( 3.5sqrt (2))^2  ( 5  - 3.5sqrt (2) )  ] 

 

3.5  units^2

 

 

cool cool cool

 Dec 31, 2020

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