Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | January | Q#3

 

Question

The line AB has equation  . The point C has coordinates (2,-7).

a.    

                           i.       Find the gradient of AB.

                         ii.       The line which passes through C and which is parallel to AB crosses the y-axis at the  point D. Find the y-coordinate of D.

b.   The line with equation  intersects the line AB at the point A. Find     the coordinates of  A.

c.    The point E has coordinates (5,k). Given that CE has length 5, find the two possible values of the constant k.

Solution

a.
 

                            i.
 

We are given equation of the line;

We are required to find the gradient (slope) of the line.

Slope-Intercept form of the equation of the line;

Where  is the slope of the line.

We can rearrange the given equation in slope-intercept form as follows;

Hence equation gradient (slope) of the line is .

 

                          ii.
 

It is evident that we are required to find the coordinates of y-intercept of line, say , which passes  through point C(2,-7) and is parallel to line AB.

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 need equation of line .

To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope  form of Equation of Line).

We already have coordinates of point on the line  and that is C(2,-7).

Therefore, we need slope of line .

We know that line   is parallel to line AB.

If two lines are parallel to each other, then their slopes  and  are equal;

 Therefore;

We have found in (a:i) that;

Now we can write equation of the line .

Point-Slope form of the equation of the line is;

Now to find the coordinates of point D i.e. y-intercept of line .

Substituting  in equation of ;

Hence,

b.
 

Here, we are required to find the coordinates of point A which is intersection of of line, say , with  equation  and line AB.

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 AB as found in (a:i) is;

Equation of the line  is;

Equating both equations;

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

With x-coordinate of point of intersection of two lines (or line and the curve) at hand, we can find the  y-coordinate of the point of intersection of two lines (or line and the curve) by substituting value  of x-coordinate of the point of intersection in any of the two equations.

We choose;

Therefore, coordinates of point A are (-1,5).

c.
 

Expression to find distance between two given points  and is:

Therefore;

We are given the coordinates of both points as C(2,-7) and E(5,k) and we are also given that  .

Hence;

 

Now we have two options.

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