# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#7

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Question

The line  has equation . The line  passes through the point  and is perpendicular to .

i.
Find the equation of .

ii.       Given that the lines  and  intersect at the point B, find the length of AB.

Solution

i.

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).

In the given case, to find the equation of  we have coordinates of one point on the line . Therefore we need slope of the line  to write its equation.

We can find the slope of the line   from the fact that line  and line  are perpendicular.

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

Therefore for the given case;

Slope-Intercept form of the equation of the line;

Given equation of the line ;

Therefore;

Hence slope of  can be found;

Now we have requisite information to write the equation of the line  i.e. coordinates of point and slope.

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

ii.

To find the length of line ;

To find the length of line, we need coordinates of both end-points of the line.

We have coordinates of one point of the line AB i.e. . To find the coordinates of the other point of the line AB i.e. B we can utilize the fact that lines  and  intersect at the point B.

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate  coordinates of both lines i.e. equate equations of both the lines (or the line and the
curve).

Since B is the point of intersection of both line, therefore, it lies on both lines. Hence  and  coordinates of point B have same values on both lines. Therefore, we can equate  coordinates of both lines i.e. equate equations of both lines.

Equation of line .

Equation of line .

Equating both equations;

Hence the x-coordinate of point B is 3. To find the y-coordinate of the point B we can substitute x-coordinate of point B in any of the two equations of lines  and  because point B is the point of intersection of both lines i.e. it lies on both lines.

We choose equation of line ;

Substituting

Hence the coordinates of the point .

Now that we have both points  and  , we can write the equation of the line AB.

Expression to find distance between two given points  and  is:

In the given case;  and