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
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