# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2015 | June | Q#1

**Question**

The line AB has equation .

**a.
**

** i. **Find the gradient of AB.

** ii. **Find an equation of the line that is perpendicular to the line AB and which passes through the point (-2,-3) . Express your answer in the form , where p, q and r are integers.

**b. **The line AC has equation . Find the coordinates of A.

**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;

**b. **

We are required to find the equation of line L.

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 point on L line (-2,-3).

Therefore, we need slope of the line L which is perpendicular to line AB.

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

Therefore;

From (a) we have slope of line AB as;

Hence;

Now with coordinates of point (-2,-3) on the line L and slope of the same , we can write equation of line L.

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

**c.
**

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

We are also given equation of line AB as .

It is evident that point A 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 3;

Equation of the curve is;

Multiplying both sides by 5;

Adding both equations;

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;

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