Question

A circle with centre C has . 

a.   Express this equation in the form

b.   Show that the circle touches the y-axis and crosses the x-axis in two distinct points.

c.   A line has the equation , where  is a constant.

                           i.       Show that the x-coordinates of any points of intersection of the circle and the line satisfy  the equation

.

                         ii.       Hence, find the value of k for which the line is a tangent to the circle.

Solution

a.
 

We are given equation of the circle with center C as;

Expression for a circle with center at  and radius  is;

We can write the given equation of the circle in standard for as follows.

We have the algebraic formula;

For the given case we can rearrange the given equation and compare the given terms with the  formula.

For terms containing	For terms containing
Therefore, we can deduce that;

To complete the square we can add and subtract the deduced value of ;

b.
 

We are required to show that there is just a single y-intercept and two distinct x-intercepts of the  circle.

The point  at which curve (or line) intercepts x-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

We have found in (a) that equation of the circle can be written as;

Therefore, for coordinates of x-intercepts we substitute .

Two distinct values of x indicate that there are two intersection points.

For coordinates of y-intercepts, we substitute

Single value of x indicates that there is only one intersection point.

Hence, the given circle intersects x-axis at two distinct points whereas only single intersection point  with y-axis.

c.    

                            i.
 

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;

Substituting values of  in equation of the circle;

 

                          ii.
 

If the line is tangent to the circle then solution of equation which satisfies the x-coordinates of any  points of intersection of the line and the circle must yield a single solution/equal roots.

We have seen from (c:i) that it results in following equation.

To find the possible values of  we need to solve above equation.

For a quadratic equation , the expression for solution is;

Where  is called discriminant.

If , the equation will have two distinct roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

Since it is a quadratic equation, of it is to yield single/equal roots then its discriminant must be equal to ZERO.

For the above equation to be a quadratic equation;