When the coordinate axes are rotated by an angle tan - 1 3 4 about the origin, then the equation x 2 + y 2 = 9 is transformed to the equation

Step-by-step solution

Let θ be the angle of rotation, then θ = tan - 1 3 4 . Now, new coordinate axes are X = x cos θ - y sin θ Y = x sin θ + y cos θ Then, X = x cos tan - 1 3 4 - y sin tan - 1 3 4 Y = x sin tan - 1 3 4 + y cos tan - 1 3 4 Or X = x 4 5 - y 3 5 Y = x 3 5 + y 4 5 On solving, we get x = 4 X + 3 Y 5 y = 4 Y - 3 X 5 Now, putting these values in x 2 + y 2 = 9 , we get 4 X + 3 Y 2 25 + 4 Y - 3 X 2 25 = 9 or X 2 + Y 2 = 9