# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2016 | June | Q#1

Question

The line AB has equation .

a.   The line AB is parallel to the line with equation .  Find the value of m.

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

c.    The point with coordinates  lies on the line AB. Find the value of k.

Solution

a.

We are given equation of line AB as;

Slope-Intercept form of the equation of the line;

Where  is the slope of the line.

We can rearrange the equation of line AB in slope intercept form.

Hence , slope of line AB is;

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

We know that line with equation  is parallel to line AB, therefore;

b.

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

The line has equation  and we are required to find the coordinates of point B.

We are also given equation of line AB as .

It is evident that point B is the intersection point of both the 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;

Multiplying both sides by 2;

Equation of the curve is;

Multiplying both sides by 3;

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;

Hence, coordinates of point .

c.

If a point P(x,y) lies on a line, then its coordinates satisfy the equation of the line.

We are given that point  lies on the line AB.

We are also given equation of the line AB as;

Since point  lies on the line AB, its coordinates must satisfy equation of the line AB.

Substitute coordinates of the given point in equation of the line AB.