Question

The points P(0, 2) and Q(3, 7) lie on the line , as shown in Figure.

The line  is perpendicular to , passes through Q and crosses the x-axis at the point R, as shown in Figure.

Find

a.   an equation for , giving your answer in the form ax + by + c = 0, where a, b and c are integers,

b.   the exact coordinates of R,

c.   the exact area of the quadrilateral ORQP, where O is the origin.

Solution

a.
 

We are required to find equation of the 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 already have coordinates of a point on the line  Q(3, 7). Therefore, we need slope of the  normal to write its equation.

We are given that the line  is perpendicular to .

If a line  is normal to the curve , then product of their slopes  and  at that point (where line  is normal to the curve) is;

Therefore;

We can find slope of the line  if we have slope of the line .

We need to find the slope of the line .

Expression for slope (gradient) of a line joining points  and ;

Since line  is joined by points P(0, 2) and Q(3, 7), therefore;

Hence;

Now we can write equation of the line .

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

b.
 

We are required to fins the exact coordinates of point R.

It is evident that we are looking for the coordinates of 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 (a) as;

We substitute y=0 in this equation;

Hence, exact coordinates of point .

c.    

We are required to find the exact area of the quadrilateral ORQP, where O is the origin. 

We can spot two triangles in the given quadrilateral ORQP as shown below.

It is evident that  has  and  has .

It is evident from the diagram that;

Expression for the area of the triangle with base  and height  is;

Therefore, first we need to find distances OP, OR, QR and QP.

Expression for the distance between two given points  and is:

We have coordinates of these points as given and found as follows;

Q(3,7)

P(0,2)

O(0,0)

Hence;

Now we have;

Therefore;

Finally;