The line l is a tangent to the circle x^2 + y^2 = 40 at the point A. A is the point (2, 6). The line l crosses the x-axis at the point P. Work out the area of triangle OAP.

This will involve drawing a diagram to aid your answer. Where O is the origin, A is a point on the circle and P is a point on the x-axis. OA is the radius of the circle. Line l crosses through point A and meets the x-axis at point P. First thing to do will be to work out the gradient of OA = 3 As we know the tangent perpendicular to the radius of the circle (OA) and so the gradient of the tangent and radius must multiply to give -1, and so as the gradient of OA is 3, the gradient of the tangent is -1/3. We know that the tangent goes through the point (2,6) and so, as y - y1 = m(x-x1), the equation of the tangent is equal to y = -x/3 + 20/3 We know that P goes through the x-axis and so at P, y = 0. If y= 0 x = 20 We know that the area of the triangle is 1/2 x base x height = the base is equal to the distance between O and P (20), and the height is equal to the distance between point A and the base (6 - as point A has y coordinate 6) so... = 1/2 x 20 x 6 = 60