# 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 .