Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01R)  Year 2014  June  Q#7
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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 (TwoPoint form of Equation of Line) or coordinates of one point on the line and slope of the line (PointSlope form of Equation of Line).
We have coordinates of both points L(–1, 2) and M (7, –4).
TwoPoint 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;
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