# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#4

Question

C is the mid-point of the line joining A(14,7) to B(6,3). The line through C perpendicular to AB  crosses the y-axis at D.

i.       Find the equation of the line CD, giving your answer in the form

Solution

i.

We are required to write equation of the line CD.

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 need coordinates of a point on line CD and also slope of line CD.

Let’s find coordinates of point C.

We are given that C is the mid-point of the line joining A(14,7) and B(6,3).

To find the mid-point of a line we must have the coordinates of the end-points of the line.

Expressions for coordinates of mid-point of a line joining points  and;

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

Therefore;

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

Hence coordinates of point C are (4,-2).

Now we find slope of line CD.

We are given that line CD is perpendicular to AB.

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

Therefore, if we know slope of line AB we can find slope of line CD.

Let’s find slope of line AB with points A(14,7) and B(6,3).

Expression for slope of a line joining points  and ;

Therefore;

Hence;

Now, with coordinates of point on the line CD as C(4,-2) and slope of line CD as , we can  write equation of line CD.

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

Therefore;

ii.

We are required to find distance AD.

Expression to find distance between two given points  and is:

We already have coordinates of A(14,7). We need to find coordinates of point D.

We have found in (i) equation of line CD as;

We are also given that line CD crosses the y-axis at D which means point D is y-intercept of line CD.

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).

Therefore;

Hence, coordinates of point D are (0,-10).

Now we can find length AD.