Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/11) | Q#6

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Question

The points A(1,1) and B(5,9) lie on the curve .

     i.       Show that the equation of the perpendicular bisector of AB is 2y = 13 x.

The perpendicular bisector of AB meets the curve at C and D.

   ii.       Find, by calculation, the distance CD, giving your answer in the form  , where p and q are  integers.

Solution


i.
 

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 the perpendicular bisector of AB nor slope of the  perpendicular bisector of AB.

Let’s find both to write equation of the perpendicular bisector of AB.

Since we are looking for equation of perpendicular bisector of AB, the mid-point of AB will also lie  on perpendicular bisector of AB.

Let’s find coordinates of mid-point of AB.

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

We are given coordinates of both points A(1,1) and B(5,9).

Therefore;

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

Hence coordinates of a point P on perpendicular bisector of AB are P(3,5).

Next, we proceed to find slope of perpendicular bisector of AB.

If two lines are perpendicular (normal) to each other, then product of their slopes  and  is;

Therefore, if we know slope of AB we can find slope of perpendicular bisector of AB.

Expression for slope of a line joining points  and ;

We are given coordinates of both points A(1,1) and B(5,9).

Therefore;

 

We can now find slope of perpendicular bisector of AB.

Now we can write equation of perpendicular bisector of AB.

Point-Slope form of the equation of the line is;

   ii.
 

We are required to find the distance CD.

Expression to find distance between two given points  and is:

We do not have coordinates of both points C and D.

However, we know that points C and D are points of intersection of the curve and perpendicular bisector of AB.

Let’s find the coordinates of these points of intersection.

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 perpendicular bisector of AB as found in (i);

Equation of the curve is given as;

To equate both equations we multiply both sides of equation of the perpendicular bisector of AB with 3.

Equating both equations;

Now we have two options.

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 perpendicular bisector of AB as found in (i);

When ;

When ;

Therefore coordinates of points of intersection of curve and perpendicular bisector of line AB are  and .

 

Now we can find the distance CD.

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