Question

Given a circle: C(1,2) & radius `sqrt(5)` a) Find the perpendicular distance from center to `x + 2y -10=0`, show this line is a tangent to the circle. b) Find the perpendicular distance from center to `x+2y -12 =0`, show the line does not meet circle?

Answer 1

The distance of C to the first line is exactly `sqrt5=radius`, so it is tangent, the distance of C to the second line is `7/5sqrt(5)> sqrt5=radius` so it is external

Explanation:

The general formula for the distance of the generis point `P(x_0;y_0)` to the line `ax+by+c` is

`d=abs(ax_0+by_0+c)/sqrt(a^2+b^2)`

so the distances are

`d_1=abs(1+4-10)/sqrt5=5/sqrt5=sqrt5`

`d_2=abs(1+4-12)/sqrt5=7/sqrt5=7/5sqrt5`