If the tangent and the normal at the point ( 3 , 1) to the circle x^2+ y ^2=4 , and the X -axis form a triangle, then the area (in sq.units) of this triangle is

Step-by-step solution

The circle x^2 + y^2 = 4 has center at (0,0) with radius 2 , and the given point is P( 3 , 1) . Tangent and intercept: The tangent at P has equation x 3 + y = 4 . Setting y = 0 gives x-intercept A(4/ 3 , 0) . Normal and intercept: The normal passes through the center and P , yielding y = x/ 3 . Its x-intercept is B(0, 0) . The triangle has vertices A(4/ 3 , 0) , B(0, 0) , and P( 3 , 1) . Using base AB = 4/ 3 along the x-axis and height 1 (the y-coordinate of P ), the area is 1 2 4 3 1 = 2 3 . 2 3