Question

The line  has equation 4x + 2y – 3 = 0.

a.   Find the gradient of .

The line  is perpendicular to  and passes through the point (2,5).

b.   Find an equation of  in the form y = mx + +c, where m and c are constants.

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 coordinates of a point (2,5) on line  but not slope of 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(2,5) and its slope

 at hand, we can write equation of the line .

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