Four distinct points A(2,1) , B(5,5) , C(1,8) and D(k,k) lie on a circle. If D is distinct from A , B , and C , then 7 times the area of the triangle formed by the tangent to the circle at D and the coordinate axes is equal to

Step-by-step solution

The slopes of the line segments AB and BC are: m_ AB = 5-1 5-2 = 4 3 m_ BC = 8-5 1-5 = 3 -4 = - 3 4 Since m_ AB m_ BC = -1 , the triangle ABC is right-angled at B . The circumcircle of a right-angled triangle has its hypotenuse as the diameter. Thus, AC is the diameter of the circle. Using the diametric form, the equation of the circle is: (x-2)(x-1) + (y-1)(y-8) = 0 x^2 - 3x + 2 + y^2 - 9y + 8 = 0 x^2 + y^2 - 3x - 9y + 10 = 0 Since D(k,k) lies on this circle, we substitute x=k and y=k : k^2 + k^2 - 3k - 9k + 10 =