the external centres of similitude of the circles x^2+y^2-12x+8y+48=0 and x^2+y^2-4x+2y-4=0 divides the segment joining centers in ratio

To find the ratio in which the external centers of similitude divide the segment joining the centers of the given circles, we first need to analyze the equations of the circles. Starting with the first circle:

First Circle Analysis

The equation is given as:

x² + y² - 12x + 8y + 48 = 0.

We can rewrite this in standard form by completing the square. The x terms are:

• x² - 12x → (x - 6)² - 36

And the y terms are:

• y² + 8y → (y + 4)² - 16

Substituting these back into the equation gives us:

(x - 6)² + (y + 4)² - 36 - 16 + 48 = 0.

So, we have:

(x - 6)² + (y + 4)² = 4.

This means the center of the first circle is (6, -4) and the radius is 2.

Second Circle Analysis

Next, we analyze the second circle given by:

x² + y² - 4x + 2y - 4 = 0.

Similarly, we complete the square for the x and y terms:

• x² - 4x → (x - 2)² - 4
• y² + 2y → (y + 1)² - 1

Substituting back, we have:

(x - 2)² + (y + 1)² - 4 - 1 - 4 = 0.

This simplifies to:

(x - 2)² + (y + 1)² = 9.

The center of the second circle is (2, -1) and the radius is 3.

Finding the Centers and the External Centers of Similitude

Now that we have the centers, we find the external center of similitude. The centers of the circles are:

• Circle 1: (6, -4)
• Circle 2: (2, -1)

The distance between the centers can be calculated using the distance formula:

d = √[(x₂ - x₁)² + (y₂ - y₁)²].

Plugging in our values:

d = √[(2 - 6)² + (-1 + 4)²] = √[16 + 9] = √25 = 5.

Ratio of the Division by External Centers of Similitude

To find the ratio in which the external centers of similitude divide the segment joining the centers, we use the formula:

Ratio = r₁ : r₂, where r₁ and r₂ are the radii of the two circles.

Here, r₁ = 2 and r₂ = 3. Therefore, the ratio is:

Ratio = 2 : 3.

Final Thoughts

The external centers of similitude thus divide the segment joining the centers of the circles in the ratio of 2:3. This means that if you were to draw a line segment between the centers of the circles, the external center of similitude would be located closer to the center of the first circle (the one with the smaller radius) at two parts to three parts from the first to the second circle.