If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2 + y^2 + 4x - 6y = 12$ externally at the point $(1, -1)$, then the radius of $C$ is :

Question

The area bounded by $ y - 1 = |x| $ and $ y + 1 = |x| $ is: (a) $ \frac{1}{2} $

Answer 1

To find the radius of the circle $C$ that passes through the point $(4,0)$ and touches another circle externally, we need to consider the properties of circles and the geometry involved.

The given circle equation is $x^2 + y^2 + 4x - 6y = 12$. We can rewrite this equation to find its center and radius by completing the square:

The equation can be rewritten as:

$(x^2 + 4x) + (y^2 - 6y) = 12$

Completing the square:

$(x+2)^2 - 4 + (y-3)^2 - 9 = 12$

Simplifying this, we get:

$(x+2)^2 + (y-3)^2 = 25$

The center of the given circle is $(-2, 3)$ and its radius is $5$.

Now, let's consider the conditions for our circle $C$. It touches the given circle externally at $(1, -1)$ and passes through $(4, 0)$.

Since the two circles touch externally at $(1, -1)$, the distance between their centers is equal to the sum of their radii. Let the center of circle $C$ be $(h, k)$.

The center of $C$ lies on the line joining center $(-2,3)$ and point $(4,0)$. Given that it touches the external circle at $(1, -1)$, this point serves as the center of $C$.

The distance between centre $(1, -1)$ and (-2, 3)$ is calculated as follows:

\[ \text{Distance} = \sqrt{(-2 - 1)^2 + (3 + 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

The radius of circle $C$ is $5$, which matches the option.

Thus, the correct answer is $5$.

Answer 2