Question

The curve  intersects the line  at two points. Find the distance between the two points.

Solution

Expression to find distance between two given points  and is:

So first we need to find the coordinates of points of intersection of the curve and the line.

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).

In this case, equation of the curve is;

We can rewrite it as;

Equation of the line is;

We can rewrite it as;

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 the equation of line;

We can rewrite it as;

Hence the two points of intersection of the curve and the line are  and .

Now we can find the distance between these two points;