Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | June | Q#3
Question
The points P and Q have coordinates (–1, 6) and (9, 0) respectively.
The line is perpendicular to PQ and passes through the mid-point of PQ.
Find an equation for , giving your answer in the form ax + by + c = 0, where a, b and c are integers.
Solution
We are required to find equation of line .
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 neither coordinates of a point on line nor slope of the line
.
First we find coordinates of a point on line .
We are given the line passes through the mid-point of the line PQ and is also perpendicular to PQ.
We first find coordinates of the mid-point of line PQ.
We are given that points P and Q have coordinates (–1, 6) and (9, 0) respectively.
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 PQ
y-coordinate of mid-point of the PQ
Hence, coordinates of mid-point of line PQ are (4,3).
Next, we find slope of line .
We are given the line is perpendicular to PQ.
If a line is normal to the curve
, then product of their slopes
and
at that point (where line is normal to the curve) is;
Therefore;
Hence, if we have slope of the line PQ we can find slope of the line .
We can find slope of line PQ as follows.
Expression for slope (gradient) of a line joining points and
;
We are given that points P and Q have coordinates (–1, 6) and (9, 0) respectively.
Therefore;
Hence, gradient of the line ;
With coordinates of a point M (4,3) on the line and slope of the line
at hand , we can write equation of this line.
Point-Slope form of the equation of the line is;
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