Let A B be a line segment with mid-point C and D be the mid-point of A C . Let C 1 be the circle with diameter A B and C 2 be the circle with diameter A C . Let E be a point on C 1 such that E C is perpendicular to A B . Let F be a point on C 2 such that D F is perpendicular to A B and E and F lie on opposite sides of A B . Then, the value of sin ∠ F E C is

Step-by-step solution

According to given informations in the question, if radius of circle C 1 is 2 a , centre c is at origin 0 , 0 and A B is along X -axis, then B 2 a , 0 , A - 2 a , 0 , E 0 , 2 a and F - a , - a . Let ∠ F E C = θ Then, slope of line E F = m = tan π 2 - θ ⇒ cot θ = 2 a - - a 0 - - a = 3 ∵ m = y 2 - y 1 x 2 - x 1 ∴ sin ∠ F E C = sin θ = 1 1 + cot 2 θ = 1 1 + 9 = 1 10