Question

A circle has centre (1, 2) and radius √5. 

a. Find the perpendicular distance from the centre of the circle to the line with equation x + 2y − 10 = 0 and hence show this line is a tangent to the circle. 

b. Find the perpendicular distance from the centre of the circle to the line with equation x + 2y − 12 = 0 and hence show the line does not meet the circle.

Answer 1

The perpendicular distance of (1, 2) from the line x + 2y −10 = 0 is

|1 + 4 - 12|/√(1² + 2²) = √5

which equals the radius. Therefore the line is tangent to the circle. 

b The perpendicular distance of (1, 2) from the line x +2y −12 = 0 is

|1 + 4 - 12|/√(1² + 2²) = 7/√5 = 7√5/5,

which is greater than the radius √5. Hence the line does not touch the circle.