# 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?

Nov 14, 2016

The distance of C to the first line is exactly $\sqrt{5} = r a \mathrm{di} u s$, so it is tangent, the distance of C to the second line is $\frac{7}{5} \sqrt{5} > \sqrt{5} = r a \mathrm{di} u s$ so it is external

#### Explanation:

The general formula for the distance of the generis point P(x_0;y_0) to the line $a x + b y + c$ is

$d = \frac{\left\mid a {x}_{0} + b {y}_{0} + c \right\mid}{\sqrt{{a}^{2} + {b}^{2}}}$

so the distances are

${d}_{1} = \frac{\left\mid 1 + 4 - 10 \right\mid}{\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5}$

${d}_{2} = \frac{\left\mid 1 + 4 - 12 \right\mid}{\sqrt{5}} = \frac{7}{\sqrt{5}} = \frac{7}{5} \sqrt{5}$