Points P - 3 , 2 , Q 9 , 10 and R α , 4 lie on a circle C with P R as its diameter. The tangents to C at the points Q and R intersect at the point S . If S lies on the line 2 x - k y = 1 , then k is equal to _____ .

Step-by-step solution

Let S ≡ x 1 , y 1 Given: P ≡ - 3 , 2 Q ≡ 9 , 10 R ≡ α , 4 Since, P Q ⊥ R Q , so m P Q × m Q R = - 1 ⇒ 10 - 2 9 + 3 × 10 - 4 9 - α = - 1 ⇒ 1 9 - α = - 1 4 ⇒ α = 13 So, R ≡ 13 , 4 So, centre of circle is O ≡ 13 - 3 2 , 4 + 2 2 ≡ 5 , 3 Also, m O Q · m Q S = - 1 ⇒ 3 - 10 5 - 9 m Q S = - 1 ⇒ m Q S = - 4 7 Equation of Q S is y - 10 = - 4 7 x - 9 ⇒ 4 x + 7 y = 106