A differentiation problem in parametric context with solution

This is hot off the press-a question taken from the recently concluded 2011 September Preliminary Examinations of a school in Singapore. It deals with applications of differentiation in the parametric context. Extensive trigonometry is employed here together with the manipulation of surd forms. I have personally worked out everything for your (the student’s) reference.

If you want a real calculus challenge, the problem below should satisfy your appetite. Peace.

QUESTION :

The parametric equations of a curve  are

and

where is a positive constant

(i) Find the equation of the tangent to the curve at the point P  where .

(ii) The normal to the curve at the point where intersects the axis at . Find the coordinates of and hence show that the area enclosed by the normal at , the tangent and the x-axis is

Author

Frederick Koh is a teacher residing in Singapore who specialises in teaching the A level maths curriculum. He has accumulated more than a decade of tutoring experience and loves to share his passion for mathematics on his personal site www.whitegroupmaths.com.

Mr Koh is also the author of the post Working with summation.

I have created a GeoGebra applet to visualize Question 1 above.