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

**Question**

The points A and B have coordinates (1,6) and (5,-2) respectively. The mid-point of AB is M.

**a. **Find the coordinates of M.

**b. **Find the gradient of AB, giving your answer in its simplest form.

**c. **A straight line passes through M and is perpendicular to AB.

** i. **Show that this line has equation .

** ii. **Given that this line passes through the point (k, k+5) , find the value of the constant k.

**Solution**

**a.
**

We are given points A and B with coordinates (1,6) and (5,-2) respectively and required to find the coordinates of their mid-point of M.

To find the mid-point of a line we must have the coordinates of the end-points of the line.

Expressions for coordinates of mid-point of a line joining points and;

x-coordinate of mid-point of the line

y-coordinate of mid-point of the line

Therefore, for the given case;

x-coordinate of mid-point of the line

y-coordinate of mid-point of the line

Hence, coordinates of mid-point (M) of AB are (3,2).

**b. **

We are given points A and B with coordinates (1,6) and (5,-2) respectively and required to find the gradient of line AB.

Expression for slope (gradient) of a line joining points and ;

Therefore, for the given case;

**c. **

** i.
**

We are given that a straight line, say , passes through M and is perpendicular to line AB.

We are required to write 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).

From (a) we have coordinates of point M(3,2) but we need to find the slope of 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 (b) we have slope of the line AB as .

Hence;

Now with the coordinates of a point on line as M(3,2) and slope of the line as , we can write equation of line .

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

ii.

We are given that line passes through a point (k,k+5).

If a line passes through a point, then coordinates of this point must satisfy the equation of the line.

From (c:i) we have found equation of line ;

Therefore, for point (k,k+5), equation must satisfy.

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