Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | 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 x-axis 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 (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 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 .
Point-Slope 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 x-intercept of the line .
The point at which curve (or line) intercepts x-axis, 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;
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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) |
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Hence;
Now we have;
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Therefore;
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Finally;
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