# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2009 | June | Q#8

Question

The points A and B have coordinates (6, 7) and (8, 2) respectively.

The line  passes through the point A and is perpendicular to the line AB, as shown in Figure 1.

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

Given that  intersects the y-axis at the point C, find

b)  the coordinates of C,

c)   the area of ΔOCB, where O is the origin.

Solution

a)

We are required to find equation of .

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 are given that line  passes through the point A (6, 7) and is perpendicular to line AB.

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, if we have slope of line AB we can find slope of line .

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

We have coordinates of both points A and B as (6, 7) and (8, 2) respectively

Hence;

Therefore;

Now we can write the equation of line .

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

Therefore;

b)

We are given that line  intersects x-axis at point C, therefore, C is the y-intercept of line

We are required to find the coordinates of point C ie y-intercept.

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 have found equation of line  in (a);

Substitution of x=0;

Hence, coordinates of point C.

c)

We are required to find the area of triangle OCB.

Expression for the area of the triangle is;

We need both base and height of triangle OCB to find area.

We need to locate the points O, B and C to sketch the triangle OCB.

We have coordinates of both points A and C as (6, 7) and C respectively.

We are given that point O is origin therefore O(0,0).

We locate points O, B and C and sketch the triangle OCB as shown below.

It is evident from the diagram that;

Base of Triangle OCB=OC

Height of Triangle OCB =QB

We need to find these lengths to calculate area of triangle OCB.

Base of Triangle OCB=OC=Distance from O to C along Y-axis=

Height of Triangle OCB =QB=Distance from Q to B along X-axis=8

Now we can find area of triangle OCB.