Question

The line $x + y = 2$ touches a circle. If the centre of the circle is at $(-4,\,0)$, then the radius of the circle is

• A. $2\sqrt{2}$
• B. $\dfrac{3\sqrt{2}}{2}$
• C. $\sqrt{2}$
• D. $\dfrac{\sqrt{2}}{2}$
• E. $3\sqrt{2}$

Hint:

The perpendicular distance from point $(x_0,y_0)$ to line $ax+by+c=0$ is $\dfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$. This equals the radius when the line is tangent to the circle.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
If a line is tangent to a circle, the radius equals the perpendicular distance from the centre to the line.

Step 2: Detailed Explanation:
Line: $x + y - 2 = 0$. Centre: $(-4,\,0)$.
Perpendicular distance $= \dfrac{|(-4) + 0 - 2|}{\sqrt{1^2+1^2}} = \dfrac{|-6|}{\sqrt{2}} = \dfrac{6}{\sqrt{2}} = 3\sqrt{2}$.

Step 3: Final Answer:
Radius $= 3\sqrt{2}$.