What ratio does the point (6, 6) divide the line segment joining the centres of circles C₁ and C₂?

Steps to Solve

1. Identify the coordinates of the circle centers

Let the coordinates of the first circle center be $(x_1, y_1)$ and the second circle center be $(x_2, y_2)$. For example, if the coordinates are $(2, 3)$ and $(4, 7)$, then $x_1 = 2$, $y_1 = 3$, $x_2 = 4$, and $y_2 = 7$.

2. Determine the coordinates of the point that divides the segment

Let the point dividing the segment be $(x, y)$. For the example, let's take $(3, 5)$, which is the point we will use in our calculations.

3. Use the section formula

According to the section formula, if a point divides the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$, the coordinates of the point are given by:

$$ x = \frac{mx_2 + nx_1}{m+n} $$ $$ y = \frac{my_2 + ny_1}{m+n} $$

4. Set up equations with the known coordinates

From our points:

For $x$ coordinate: $$ 3 = \frac{m \cdot 4 + n \cdot 2}{m+n} $$

For $y$ coordinate: $$ 5 = \frac{m \cdot 7 + n \cdot 3}{m+n} $$

5. Cross-multiply to eliminate fractions

For the $x$ coordinate: $$ 3(m+n) = 4m + 2n $$ Expanding gives: $$ 3m + 3n = 4m + 2n $$ Rearranging leads to: $$ m - n = 3n $$ or $$ m = 4n $$

For the $y$ coordinate: $$ 5(m+n) = 7m + 3n $$ Expanding gives: $$ 5m + 5n = 7m + 3n $$ Rearranging leads to: $$ 2m = 2n $$ which simplifies to: $$ m = n $$

6. Set up the ratio

Using $m = 4n$ and $m = n$, we substitute $n$ from the second equation into the first: $$ n = 4n $$ Solving gives: $$ 4n = n $$ This leads to the conclusion that $m = 4$ and $n = 1$ forming the ratio.

7. Write the final ratio

Thus the point divides the line segment joining the two circle centers in the ratio of $4:1$.