If x = 2 n π + t a n - 1 p q and y = r is a solution of the equation 12 sin ⁡ x + 5 cos ⁡ x = 2 y 2 - 8 y + 21 , then the value of k , such that p 2 + q 2 + k r 2 = 15 , is equal to

Step-by-step solution

12 sin ⁡ x + 5 cos ⁡ x = 2 y 2 - 8 y + 2 1 12 sin ⁡ x + 5 cos ⁡ x lies between - 13,13 Also, 2 y 2 - 8 y + 21 = 2 y 2 - 4 y + 4 + 13 = 2 y - 2 2 + 13 So, possibility of equality is when LHS = RHS = 13 RHS = 13 , when y = 2 ⇒ r = 2 . 12 sin ⁡ x + 5 cos ⁡ x = 13 ⇒ sin ⁡ x 12 13 + cos ⁡ x 5 13 = 1 ⇒ s i n x + c o s - 1 12 13 = 1 ⇒ x + c o s - 1 12 13 = 4 n + 1 π 2 ⇒ x = 2 n π + π 2 - c o s - 1 12 13 = 2 n π + s i n - 1 12