# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2005 | January | Q#3

Question

A circle has equation

a.   By completing  the square, express the equation in the form

b.   Write down:

i.       the coordinates of the center of the circle;

ii.       the radius of the circle

c.   The line with equation  intersects the circle at the points P and Q.

i.       Show that the x-coordinates of P and Q satisfy the equation

ii.       Find the coordinates of P and Q.

Solution

a.

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.

If a circle is given by the equation  then the center of circle is at   and  is the radius of the circle.

Therefore, for the given case;

i.

Coordinates of center of circle .

ii.

c.

Equation of the line given is;

Since line and the circle intersect, we find the coordinates of points of intersection.

i.

If equations of line and the circle are given;

ü From equation of line make  subject.

ü Substitute this into equation of the circle to get a quadratic equation in terms of

ü To find the coordinates of points of intersection solve the quadratic equation for  and  substitute the found values of  into the equation of line to find  corresponding values of .

Equation of the line is;

Equation of the circle is;

Substituting value of  from equation of line into equation of circle;

ii.

We continue from (c:i) to find the coordinates of P and Q.

We solve the equation;

Now we have two options.

Two roots indicate that there are two intersection points.

We choose equation of the line;

 For For

Hence coordinates of  and .