# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | January | Q#2

**Question**

The point has coordinates and the point has coordinates .

The line has equation .

**a. **

** i. **Show that .

** ii. **Hence find the coordinates of the mid-point of .

**b. **Find the gradient of .

**c. **Line is perpendicular to the line .

** i. **Find the gradient of .

** ii. **Hence find the equation of the line .

** iii. **Given that point lies on the x-axis, find its x-coordinate.

**Solution**

**a.
**

** i.
**

We are given that point lies on the line whose equation is given as;

Since coordinates of a point on the given line must satisfy equation of the lien.

Therefore must satisfy the equation;

** ii.
**

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

For the given case we have the point ( and the point . Therefore;

x-coordinate of mid-point of the line

y-coordinate of mid-point of the line

Hence mid-point of has coordinates .

**b. **

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

For the given case we have the point ( and the point . Therefore;

**c. **

** i.
**

We are given that line is perpendicular to the line .

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

Therefore;

From (b) we have ;

** ii.
**

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).

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

For line AC we have coordinates one point on the line and also slope from (c:i) as .

Therefore;

** iii.
**

We are given that point lies on x-axis that means it is x-intercept of line .

The point at which curve (or line) intercepts x-axis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).

Equation of the line , from (c:ii) is;

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