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

Question

A circle has center  and radius 5. The point  has coordinates .

a.   Write down the equation of the circle.

b.   Verify that point  lies on the circle.

c.   Find the gradient of the line .

d.

i.  Find the gradient of the line which is perpendicular to .

ii. Hence find an equation for the tangent to the circle at point .

Solution

a.

Expression for a circle with center at  and radius  is;

Therefore for a circle with center  and radius 5;

b.

If a point lies on a circle then coordinates of point must satisfy the equation of the given circle.

From (a) we have equation of the circle;

The point  has coordinates  ۔

Hence, substituting  and  coordinates of point  in equation of the circle;

Therefore, point  lies on the circle.

c.

Expression for slope (gradient) of a line joining points  and ;

For the given case we have  and . Therefore;

d.

i.

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

Therefore;

From (C) we have . Hence;

ii.

If we are required to find the equation of tangent to a circle at point :

ü Find the center  of the circle