Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2009 | June | Q#1

 

Question

The line AB has equation .

a.   

                    i.       Find the gradient of AB.

                  ii.       The point A has coordinates (2,1) . Find an equation of the line which passes through the  point A and which is perpendicular to AB.

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

Solution

a.
 


i.
 

We are given equation of the line AB;

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

Slope-Intercept form of the equation of the line;

where  is the slope of the line.

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

Hence, gradient of AB is .


ii.
 

We are required to find the equation of line  which passes through a point A(2,1) and is  perpendicular to line AB.

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 a point on line  as A(2,1). Therefore, we need slope of the line .

We are given that line  is perpendicular to line AB.

If two lines are perpendicular (normal) to each other, then product of their slopes  and  is;

Therefore;

From (a:i) we have;

Hence;

Now we can write the equation of line  with the help of coordinates of point A(2,1) on the line and  slope of the line .

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

b.
 

We are required to find the coordinates of point of intersection of two lines. 

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 is;

We have rearranged the equation in (a:i) as;

Equation of the other line is;

We can rearrange the equation as;

Equating both equations of lines;

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;

Substituting ;

Hence coordinates of point C(7,-2).

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