# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2014 | June | Q#9

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**Question**

The line , shown in figure has equation 2x+3y = 26.

The line

**a. **Find an equation for the line

The line

Line

**b. **Find the area of triangle OBC.

Give your answer in the form

**Solution**

**a. **

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 coordinates of a point O(0,0) on 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

We need 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

Therefore;

With coordinates of a point on the line

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

**b. **

We are required to find area of triangle OBC.

Expression for the area of the triangle with base

Consider the figure below.

It is evident that OB can be considered as base of triangle OBC whereas AC is height of the triangle.

We need to find distances OB and AC.

Expression for the distance between two given points

We do not have coordinates of these points so let us find them first.

We have coordinates of O(0,0).

We need to find coordinates of point B, next.

Point B is the y-intercept of the line

The point

We substitute x=0 in equation of the line

Hence, coordinates of y-intercept of the line

Next we need coordinates of point C which is the point of intersection of 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

Equation of the line

Equation of the line

We need to solve these simultaneous equations.

Substitute expression of y from second equation in first;

.

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 equation of the line

Substitute

Hence, coordinates of point of intersection of lines C are (4,6).

Now we can find distances OB and AC.

First we can see for OB that it is only vertical (along y-axis) distance of point B

Similarly, AC is only horizontal (along x-axis) distance of point C(4,6) from point A, therefore;

Hence;

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