Question

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

a.   Find the gradient of .

The line  crosses the x-axis at the point A and the y-axis at the point B, as shown in Figure.

The line  is perpendicular to  and passes through B.

b.   Find an equation of .

The line  crosses the x-axis at the point C.

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  is the slope of the line.

Therefore, we can rearrange the given equation of line  in slope-intercept form, as follows, to find  the gradient of the line.

Hence, gradient of the line  is;

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  nor slope of the line .

First we find coordinates of a point on line .

We are given that line  passes through B. We are also given that the line  crosses the y-axis at  the point B. Therefore, point B is y-intercept of the line .

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

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  to write its equation.

We are given that the line  is perpendicular to .

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

Therefore, if we have slope of line , we can find slope of the line .

From (a), we have found that , therefore;

With coordinates of a point on the line  as B(0,4) and its slope

 at hand, we can write equation of 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  crosses the x-axis at the point A. Therefore, point A is the x-intercept of  the 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).

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  crosses the x-axis at the point C. Therefore, point C is the x-intercept of  the 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).

We have found equation of the line  in (b);

We substitute ;

Therefore, coordinates of point C are .

We sketch  triangle with the help of these coordinates of three points.

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  and is:

We have coordinates of  and . Therefore;

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;