[H2_20150512PCD] Quadratic Discriminant for a Parametric Curve

Question

Introduction

     To begin with, do you notice that there are many letters (in italics) in the above?  There seems to be a confusing mix of variables and constants.  There are only three variables:  x,  y,  and  t.  The constants are  a,  s,  and  p.  There is another letter ‘l’,  which is the name/label for a straight line.  It is good to highlight or mentally mark these different things as different.

     Since the topic is on parametric differentiation, it seems that you need to use differentiation to solve this question.  Notice that the equations involved are at worst quadratic?  No square roots, cosines, logarithms, cubes, exponentials ... etc.  Whilst it is not wrong to use differentiation, there is a slick way – using quadratic discriminants.  In fact, this is the first thing you should think of if you see that the equations involved link to a quadratic equation.

Solution

Remarks

     We should always make it a habit to check and justify division by zero.  It is dangerous to divide an equation throughout by a variable or constant if you do not know what it is, or whether it is zero.  Make sure it is not zero before dividing.

     The quadratic discriminant method cannot be used unless you have things that reduce to quadratic equations.  But when it can be used, it is very powerful and it gives a direct answer.  Note that here we are not solving for the variable  t.  We are solving for the constant  s  in the first part, and for the constant  p  in the second part. 

Heuristics Used

H04. Look for pattern(s)

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Levels, H2 Mathematics 

* International Baccalaureate Mathematics 

* other syllabuses that involve quadratic discriminants