# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2012 | January | Q#6

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**Question**

The line has equation 2x − 3y +12 = 0.

**a. **Find the gradient of

The line

The line

**b. **Find an equation of

The line

**c. **Find the area of triangle ABC.

**Solution**

**a. **

We are given equation of line

We are required to find the gradient of

Slope-Intercept form of the equation of the line;

Where

Therefore, we can rearrange the given equation of line

Hence, gradient of the line

**b. **

We are required to find 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).

We have neither coordinates of a point on line

First we find coordinates of a point on line

We are given that line

The point

We are given equation of line

We substitute

Hence, coordinates of point B are (0,4).

Now we have coordinates of a point on the line

Next we need slope of the line

We are given that the line

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

Therefore, if we have slope of line

From (a), we have found that

With coordinates of a point on the line

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

**c.
**

We are required to find the area of

We need coordinates of all three vertexes of triangle to sketch it in order to calculate its area.

We have found in (b) that coordinates of point B are (0,4).

We now find coordinates of point A.

We are given that the line

The point

We are given equation of the line

We substitute

Therefore, coordinates of point A are (-6,0).

Next, we find coordinates of point C.

We are given that the line

The point

We have found equation of the line

We substitute

Therefore, coordinates of point C are

We sketch

Expression for the area of the triangle is;

For

We need to find AC and BD.

First we find AC.

Expression for the distance between two given points

We have coordinates of

Next we need to find BD. It is evident from the diagram that it is only the distance of B from origin D(0,0) along y-axis which is 4 since B(4,0).

Therefore;

Hence;

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