Functions (Part 1)

function,AS Exam,CIE,completing the square,domain,range,inverse function,tangent,discriminant,line,curve

This item is taken from Cambridge International AS and A Level Mathematics (9709) Pure Mathematics 1 Paper 13 of May/June 2010.
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To deal with this item, let us summarize the details of the problem:
(a) Syllabus area: Functions
(b) Formula/Concept needed:
> completed square form of quadratics
> discriminant (b^2 - 4ac)
> domain and range of functions
> inverse of a function

For the first item, let us encircle the clue words/phrase.

Since we are dealing with a line and a curve, the phrase tangent to the curve suggests the following:

function,AS Exam,CIE,completing the square,domain,range,inverse function,tangent line,line,curve

Let us start by writing the equations of the line and the curve in a function of y.

We may now use elimination or substitution method to combine the two equations. Since both are equated to y, it is easier for us to substitute the second equation to the first equation and get

Rearrange the terms on one side of the equation.

Combine like terms and simplify.

Notice that kx - 8x were combined into one (k - 8)x. Our goal to make the coefficient of x as one. Hence, the coefficient of x is k - 8.

In this form, we can now determine the values of a, b and c for the discriminant.
a = 2, b = k - 8 and c = 2

Let us now find the discriminant. Since the line is tangent to the curve, there is only one intersection. It also means that that there are two equal roots of the equation. That is

function,AS Exam,CIE,completing the square,domain,range,inverse function,tangent line,discriminant,line,curve

substituting the values of a, b and c:

Expand and simplify:

This quadratic equation can be solved using factoring or the quadratic formula. However, let us use the factoring here.

Solving for the values of k.

Hence, the values of k for which the line y + kx = 12 is tangent to the curve are 4 and 12.

[End of part 1]
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