The line y = (a^2)x and the curve y = x(b − x)^2, where 0<a<b , intersect at the origin O and at points P and Q. Find the coordinates of P and Q, where P<Q, and sketch the line and the curve on the same axes. Find the tangent at the point P.

Firstly, for the points of intersection we need to equate the two expressions for y. Since we know that they intersect at the origin, we can immediately cancel the x values and then solve the quadratic for the remaining points. The first line (y=(a^2)x) is a line through the origin and the second is a cubic, also through the origin. The repeated factor in y = x(b – x)^2 indicates tangency at x = b so we have a positive cubic that goes through the origin and then just touches the x axis at x=b. Next, for the equation of the tangent, we need to differentiate, and then substitute our value of x in order to get the gradient of the tangent at this point. Using this gradient and the values of x and y at the point P, we can then calculate the y-intercept and therefore the equation of our tangent line.