Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | 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 (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.
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We can equate both equations;
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