# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2005 | January | Q#1

Question

The point has coordinates and the point has coordinates .

a.

i.    Find the gradient of .

ii.    Hence, or otherwise, show that the line has equation .

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

Solution

a.

i.

Expression for slope (gradient) of a line joining points and ; For the given case;   ii.

Point-Slope form of the equation of the line is; For the given case;         b.

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, from (a:i) is;   Equation of the other line is;     Equating both equations;         Single value of y indicates that there is only one intersection point.

Corresponding values of x coordinate can be found by substituting values of y in any of the two  equation i.e either equation of the line or equation of the curve.

We choose;   Hence coordinates of .