Question

Let the centre of a circle, passing through the point \((0, 0)\), \((1, 0)\) and touching the circle \(x^2 + y^2 = 9\), be \((h, k)\). Then for all possible values of the coordinates of the centre \((h, k)\), \(4(h^2 + k^2)\) is equal to __________.

Answer 1

Correct Answer: 9

Given expression:

\[ |3 \, \text{adj}(2 \, \text{adj}(|A|A))| \]

Step-by-step simplification:

\[ |3 \, \text{adj}(2|A|^2 \, \text{adj}A)| \]

\[ |3 \cdot 2^2 \cdot |A|^4 \, \text{adj}(\text{adj}A)| \]

Using the property \(|\text{adj}A| = |A|^{n-1}\), we substitute and simplify:

\[ 3^3 \cdot 2^6 \cdot |A|^{12} \cdot |A|^2 = 3^{-13} \cdot 2^{-10} \]

This leads to:

\[ |A|^{16} = 3^{-16} \cdot 2^{-16} \]

Therefore,

\[ |A| = \frac{1}{6} \]

Substituting back into the main equation:

\[ |3 \, \text{adj}(2A)| = |3 \cdot 2^2 \, \text{adj}A| = 3^3 \cdot 2^6 |A|^2 \]

\[ = 3 \cdot 2^4 \]

Result:

\[ M = 4, \quad n = 1 \]

And,

\[ |3m + 2n| = 14 \]