Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2016  June  Q#10
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Question
The points P(0, 2) and Q(3, 7) lie on the line , as shown in Figure.
The line is perpendicular to , passes through Q and crosses the xaxis at the point R, as shown in Figure.
Find
a. an equation for , giving your answer in the form ax + by + c = 0, where a, b and c are integers,
b. the exact coordinates of R,
c. the exact area of the quadrilateral ORQP, where O is the origin.
Solution
a.
We are required to find equation of the line .
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 already have coordinates of a point on the line Q(3, 7). Therefore, we need slope of the normal to write its equation.
We are given that the line is perpendicular to .
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;
We can find slope of the line if we have slope of the line .
We need to find the slope of the line .
Expression for slope (gradient) of a line joining points and ;
Since line is joined by points P(0, 2) and Q(3, 7), therefore;
Hence;
Now we can write equation of the line .
PointSlope form of the equation of the line is;
b.
We are required to fins the exact coordinates of point R.
It is evident that we are looking for the coordinates of xintercept of the line .
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
We have found equation of the line in (a) as;
We substitute y=0 in this equation;
Hence, exact coordinates of point .
c.
We are required to find the exact area of the quadrilateral ORQP, where O is the origin.
We can spot two triangles in the given quadrilateral ORQP as shown below.
It is evident that has and has .
It is evident from the diagram that;
Expression for the area of the triangle with base and height is;


Therefore, first we need to find distances OP, OR, QR and QP.
Expression for the distance between two given points and is:
We have coordinates of these points as given and found as follows;
Q(3,7) 
P(0,2) 
O(0,0) 

Hence;
Now we have;




Therefore;









Finally;
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