# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | Year 2014 | June | Q#7

Question Figure 2 shows a right angled triangle LMN.

The points L and M have coordinates (–1, 2) and (7, –4) respectively.

a.   Find an equation for the straight line passing through the points L and M. Give your answer in the  form ax + by + c = 0, where a, b and c are integers.

Given that the coordinates of point N are (16, p), where p is a constant, and angle LMN = 90°,

b.   find the value of p.

Given that there is a point K such that the points L, M, N, and K form a rectangle,

c.   find the y coordinate of K.

Solution

a)

We are required to find equation of line passing through L(–1, 2) and M (7, –4).

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 both points L(–1, 2) and M (7, –4).

Two-Point form of the equation of the line is; Therefore;          b)

We are given that lines LM and MN are perpendicular.

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

Expression for slope (gradient) of a line joining points and ; Hence;  We have coordinates of both points L(–1, 2), M (7, –4) and N(16,p).        c)

We are given that there is a point K such that the points L, M, N, and K form a rectangle.

We have coordinates of both points L(–1, 2), M (7, –4), N(16,8) and K(x,y).

Expression for slope (gradient) of a line joining points and ; Slope of line LM; Slope of line MN; Slope of line KN; It is evident that line KN perpendicular to MN.

If two lines are perpendicular (normal) to each other, then product of their slopes and is;  Therefore;      It is evident that line KN must be parallel to line LM.

If two lines are parallel to each other, then their slopes and are equal; Therefore;       Now we have got following two equations and we can solve these to find y.  We can rearrange both equations for 3x.      We can equate both equations;     