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 __________.
Correct Answer: 9
Approach Solution - 1
We start with the given expression:
\[ |3 \, \text{adj}(2 \, \text{adj}(|A|A))| \]
Now simplify step-by-step:
\[ |3 \, \text{adj}(2|A|^2 \, \text{adj}A)| \]
\[ |3 \cdot 2^2 \cdot |A|^4 \, \text{adj}(\text{adj}A)| \]
We know that \(|\text{adj}A| = |A|^{n-1}\). Hence, substituting and simplifying:
\[ 3^3 \cdot 2^6 \cdot |A|^{12} \cdot |A|^2 = 3^{-13} \cdot 2^{-10} \]
Now,
\[ |A|^{16} = 3^{-16} \cdot 2^{-16} \]
\[ |A| = \frac{1}{6} \]
Substitute back into the main equation:
\[ |3 \, \text{adj}(2A)| = |3 \cdot 2^2 \, \text{adj}A| = 3^3 \cdot 2^6 |A|^2 \]
\[ = 3 \cdot 2^4 \]
Therefore,
\[ M = 4, \quad n = 1 \]
And,
\[ |3m + 2n| = 14 \]
Approach Solution -2
Step 1: General equation of the circle The equation of the circle is:
\[ (x - h)^2 + (y - k)^2 = r^2. \]
Since the circle passes through the points \((0, 0)\) and \((1, 0)\), substitute these points into the circle’s equation.
For \((0, 0)\):
\[ h^2 + k^2 = r^2. \]
For \((1, 0)\):
\[ (1 - h)^2 + k^2 = r^2. \]
Expanding and simplifying:
\[ 1 - 2h + h^2 + k^2 = h^2 + k^2. \]
Substitute \(r^2 = h^2 + k^2\) into the equation:
\[ 1 - 2h + h^2 + k^2 = h^2 + k^2. \]
Cancel \(h^2 + k^2\):
\[ 1 - 2h = 0 \implies h = \frac{1}{2}. \]
Step 2: Circle touches \(x^2 + y^2 = 9\) The given circle \(x^2 + y^2 = 9\) has a radius \(R = 3\) and is centered at \((0, 0)\). For the circle to touch \(x^2 + y^2 = 9\), the distance between their centers must be equal to the difference of their radii:
\[ \sqrt{h^2 + k^2} = R - r = 3 - \sqrt{h^2 + k^2}. \]
Let \(d = \sqrt{h^2 + k^2}\):
\[ d = 3 - d \implies 2d = 3 \implies d = \frac{3}{2}. \]
Thus:
\[ h^2 + k^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}. \]
Step 3: Compute \(4(h^2 + k^2)\) Multiply \(h^2 + k^2\) by 4:
\[ 4(h^2 + k^2) = 4 \cdot \frac{9}{4} = 9. \]
Final Answer: 9.
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