The sum of the slopes of all possible chords of the circle x 2 + y 2 = 100 , which passes through 7 , 1 and also divide the circumference of this circle into two arcs whose lengths are in the ratio 2 : 1, is equal to

Step-by-step solution

Let, the chord is A B which subtends an angle θ at the centre 0 , 0 ⇒ θ + 2 θ = 360 o ⇒ θ = 120 o = ∠ A O B Let, the distance of O from A B is equal to h Then, cos ⁡ 60 o = h 10 = 1 2 ⇒ h = 5 Let, the equation of chord is y - 1 x - 7 = m ⇒ m x - y + 1 - 7 m = 0 whose distance from 0 , 0 is equal to 5 0 - 0 + 1 - 7 m 1 + m 2 = 5 ⇒ 1 - 14 m + 49 m 2 = 25 + 25 m 2 ⇒ 24 m 2 - 14 m - 24 = 0 ⇒ m 1 + m 2 = 14 24 = 7 12