Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | Year 2013 | June | Q#11
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Question
The line meets the the curve
at the points A and B as shown in the figure.
a. Find the coordinates of A and the coordinates of B.
b. Find the distance AB in the form where r is a rational number.
Solution
a.
We are required to find the coordinates of the points of intersection of the given line and the curve.
If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate coordinates of both lines i.e. equate equations of both the lines (or the line and the curve).
Equation of the line is;
Equation of the curve is;
Substitute in equation of the curve;
Now we have two options.
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Two values of x indicate that there are two intersection points.
With x-coordinate of point of intersection of two lines (or line and the curve) at hand, we can find the y-coordinate of the point of intersection of two lines (or line and the curve) by substituting value of x-coordinate of the point of intersection in any of the two equations.
We choose equation of the line;
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Hence, the line and the curve intersect at points with coordinates and
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It is evident from the figure that point A is on the positive side of x-axis and B on the negative side of x-axis, therefore, A and B
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b.
We are required o find the distance AB.
Expression for the distance between two given points and
is:
Therefore, we need coordinates of both points A and B which we have already found in (a) as A and B
.
Hence;
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