Key Concepts and Formulas

• The line joining the centers of two circles that touch each other is perpendicular to the common tangent at the point of contact.
• If a line has slope $m$, a line perpendicular to it has slope $-1/m$.
• The parametric equation of a line passing through a point $(x_0, y_0)$ with slope $m$ is given by $x = x_0 + r\cos\theta$, $y = y_0 + r\sin\theta$, where $m = \tan\theta$ and $r$ is the distance from the point $(x_0, y_0)$.

Step-by-Step Solution

Step 1: Find the slope of the common tangent.

The equation of the common tangent is $4x + 3y = 10$, which can be rewritten as $y = -\frac{4}{3}x + \frac{10}{3}$. Therefore, the slope of the common tangent is $m_t = -\frac{4}{3}$.

Step 2: Find the slope of the line joining the centers.

Since the line joining the centers is perpendicular to the common tangent, its slope is $m_c = -\frac{1}{m_t} = -\frac{1}{-\frac{4}{3}} = \frac{3}{4}$.

Step 3: Find the equation of the line joining the centers.

The line joining the centers passes through the point of contact (1, 2) and has a slope of $\frac{3}{4}$. Therefore, its equation is $y - 2 = \frac{3}{4}(x - 1)$, which simplifies to $y = \frac{3}{4}x + \frac{5}{4}$.

Step 4: Find the coordinates of the centers using the parametric form.

Let the coordinates of the centers be $(\alpha, \beta)$ and $(\gamma, \delta)$. Since the distance from the point of contact (1, 2) to each center is equal to the radius, which is 5, we can express the coordinates of the centers using the parametric form of the line:

$$\begin{aligned} \alpha &= 1 + 5\cos\theta \\ \beta &= 2 + 5\sin\theta \\ \gamma &= 1 - 5\cos\theta \\ \delta &= 2 - 5\sin\theta \end{aligned}$$

where $\tan\theta = \frac{3}{4}$. Since $\tan\theta = \frac{3}{4}$, we can form a right-angled triangle with opposite side 3 and adjacent side 4, so the hypotenuse is $\sqrt{3^2 + 4^2} = 5$. Thus, $\sin\theta = \frac{3}{5}$ and $\cos\theta = \frac{4}{5}$.

Therefore,

$$\begin{aligned} \alpha &= 1 + 5\left(\frac{4}{5}\right) = 1 + 4 = 5 \\ \beta &= 2 + 5\left(\frac{3}{5}\right) = 2 + 3 = 5 \\ \gamma &= 1 - 5\left(\frac{4}{5}\right) = 1 - 4 = -3 \\ \delta &= 2 - 5\left(\frac{3}{5}\right) = 2 - 3 = -1 \end{aligned}$$

Step 5: Calculate $\alpha + \beta$ and $\gamma + \delta$.

We have $\alpha + \beta = 5 + 5 = 10$ and $\gamma + \delta = -3 + (-1) = -4$.

Step 6: Calculate $|(\alpha + \beta)(\gamma + \delta)|$.

$|(\alpha + \beta)(\gamma + \delta)| = |(10)(-4)| = |-40| = 40$.

Step 7: We have made an error in our logic. Instead of directly using $\theta$ and $-\theta$, we must account for the fact that the centres can be either above or below the point (1,2). Let's revisit the parametric equations:

$x = 1 \pm r\cos\theta$ $y = 2 \pm r\sin\theta$

So, the coordinates of the two centers are: $(\alpha, \beta) = (1 + 5\cos\theta, 2 + 5\sin\theta) = (1 + 5(\frac{4}{5}), 2 + 5(\frac{3}{5})) = (5, 5)$ $(\gamma, \delta) = (1 - 5\cos\theta, 2 - 5\sin\theta) = (1 - 5(\frac{4}{5}), 2 - 5(\frac{3}{5})) = (-3, -1)$

Then $\alpha + \beta = 5 + 5 = 10$ and $\gamma + \delta = -3 - 1 = -4$. Therefore, $|(\alpha + \beta)(\gamma + \delta)| = |(10)(-4)| = |-40| = 40$.

There seems to be an error in the problem or the given answer. The correct answer should be 40. Let's check the given answer again, using the fact that the distance between the centers is twice the radius.

The distance between $(\alpha, \beta)$ and $(\gamma, \delta)$ is $2r = 10$. So $\sqrt{(\alpha - \gamma)^2 + (\beta - \delta)^2} = 10$ $\sqrt{(5 - (-3))^2 + (5 - (-1))^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$. Everything seems consistent. The correct answer is indeed 40. There might be a mistake in the provided correct answer.

Common Mistakes & Tips

• Be careful with the signs when dealing with perpendicular slopes.
• Remember to consider both possible directions when using the parametric form of a line.
• Always check your calculations to avoid simple arithmetic errors.

Summary

We found the slope of the line joining the centers using the fact that it is perpendicular to the common tangent. Then we used the parametric form of the line to find the coordinates of the centers. Finally, we calculated the value of $|(\alpha + \beta)(\gamma + \delta)|$ which is 40. There appears to be an error in the provided correct answer.

Final Answer

The final answer is \boxed{40}.