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.

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;

For	For
3

Hence, the line and the curve intersect at points with coordinates  and .

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.

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;