Question

The curve   and the line  intersect at two points. Find

i.       the coordinates of the two points,

   ii.       the equation of the perpendicular bisector of the line joining the two points.

Solution

     i.
 

To find the coordinates of intersection points;

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;

Equating both equations;

Solution of Quadratic equation;

In the given case;

We have two options now;

Two values of x indicate that there are two intersection points. Corresponding values of y coordinate can be found by substituting values of x in any of the two equation i.e either equation of the line or equation of the curve.

We choose equation of the line;

For ;	For ;

Hence the two points of intersection are  and .

   ii.
 

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 are required to write the equation of the perpendicular bisector of the line joining the two points  and , therefore we first find the point on the perpendicular bisector and that is the mid-point of line joining points  and  and then find the slope of the perpendicular bisector.

To find the mid-point on the perpendicular bisector;

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

For the given case;

x-coordinate of mid-point  of
the line

y-coordinate of mid-point  of
the line

Therefore the mid-point of the line joining points  and  is;

To find the slope of the perpendicular bisector;

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

Now, to find the slope of the line joining points  and ;

Expression for slope of a line joining points  and ;

Therefore, slope of the line joining points  and ;

Hence, slope of the perpendicular bisector;

Now we can write the equation of the perpendicular bisector with point  and slope ;

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

Therefore,