Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2013 | June | Q#3

Question

A circle C has the equation a.   Write the equation of C in the form Where a, b and k are integers.

b.   Hence, for the circle C write down:

i.       the coordinates of its center;

c.

i.       Sketch the circle C.

ii.       Write down the coordinates of the point on C that is furthest away from the x-axis.

d.   Given that k has the same value as in part (a), describe geometrically the transformation which  maps the circle with equation onto the circle C.

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.

i.

Expression for a circle with center at and radius is; We can write the equation of circle obtained in (a) in standard form as; Comparing the equation from (a) with expression for circle;

i.

Coordinates of center of circle .

ii.

c.

i.

Follow following steps to sketch a circle.

ü Find the radius and coordinates of the center of the circle ü Indicate the center

ü Mark the four points which show the ends of the horizontal and vertical diameters

ü Draw the circle to pass through these four points

ü If any intercepts with the coordinate axes are integers, normally they should also be indicated ii.

Consider the sketch of the circle as shown below with slight modification. It is evident from the diagram that point P(x,y) on the circle is farthest from x-axis. The coordinates  of point P can be easily inferred from the diagram as P(5,-14).

d.

We are given that k has the same value as in part (a); Therefore, given equation will be; We have already found equation of circle C in (a) as; We are required to describe the transformation of circle equation onto .

In general a translation of transforms the graph of the circle the into the graph of The circle can be obtained from the circle by applying the translation .

Therefore;   Hence, translation vector is .