Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | Year 2014 | June | Q#9

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Question

The curve C has equation .

The line L has equation y = 3x + k, where k is a positive constant.

a.   Sketch C and L on separate diagrams, showing the coordinates of the points at which C and L cut the axes.

Given that line L is a tangent to C,

b.   find the value of k.

Solution

a)    

We are required to sketch the curve C and line L with equations given below respectively.

First we sketch the curve C with following equation.

It is evident that curve is a parabola (quadratic equation ie polynomial of degree 2).

To sketch a quadratic equation, a parabola, we need the coordinates of its vertex and x and y- intercepts, if any.

First we find the coordinates of vertex of this parabola.

Standard form of quadratic equation is;

The graph of quadratic equation is a parabola. If  (‘a’ is positive) then parabola opens upwards  and its vertex is the minimum point on the graph.
If
 (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the  graph.

We recognize that given curve , is a parabola opening upwards.

Vertex form of a quadratic equation is;

We need to rewrite the equation in vertex form as follows.

We use method of “completing square” to obtain the desired form.

Next we complete the square for the terms which involve .

Vertex form of a quadratic equation is;

Coordinates of the vertex are . Since this is a parabola opening downwards the vertex is the  maximum point on the graph. Here y-coordinate of vertex represents maximum value of  and x- coordinate of vertex represents corresponding value of .

For the given case, vertex is .

Next, we need x and y-intercepts of the parabola.

First we find the x-intercept of the parabola.

The point  at which curve (or line) intercepts x-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line). 

Therefore, we substitute  in equation of the parabola.

Since  is not possible, we can deduce that there are no x-intercepts of the parabola.

Next, we find the y-intercept of the parabola.

The point  at which curve (or line) intercepts y-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Therefore, we substitute  in equation of the parabola.

Hence, coordinates of y-intercept of the parabola are .

We can sketch the parabola as shown below.

Next we sketch the line L with following equation. 

First we find the x-intercept of the line.

The point  at which curve (or line) intercepts x-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Therefore, we substitute  in equation of the parabola.

Hence, coordinates of x-intercept of the line are .

Next, we find the y-intercept of the line.

The point  at which curve (or line) intercepts y-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line). 

Therefore, we substitute  in equation of the parabola.

Hence, coordinates of y-intercept of the parabola are .

We can sketch the line as shown below.

b)   

We are given that line L is tangent to curve C.

Therefore, line and the curve intersect at one and only one point.

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e.  coordinates of that point have same values on both lines (or on the line and the curve).  Therefore, we can equate  coordinates of both lines i.e. equate equations of both the lines (or the  line and the curve).

Equation of the line is;

Equation of the curve is;

Equating both equations;

Single value of x indicates that there is only one intersection point.

Therefore, if line L is tangent to the curve C, then this equation must yield one and only one  solution.

It is evident that it is a quadratic equation.

For a quadratic equation , the expression for solution is;

Where  is called discriminant.

If , the equation will have two distinct roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

Therefore;

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