To solve the problem step by step, we will follow the reasoning outlined in the video transcript.
### Step 1: Identify the equation of the tangent line and hyperbola
The tangent line is given by the equation:
\[ 2x - y + 1 = 0 \]
This can be rewritten in slope-intercept form as:
\[ y = 2x + 1 \]
The hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{16} = 1 \]
### Step 2: Identify the slope (m) and y-intercept (c)
From the tangent line equation \(y = 2x + 1\), we can identify:
- Slope \(m = 2\)
- y-intercept \(c = 1\)
### Step 3: Identify parameters of the hyperbola
From the hyperbola equation, we can identify:
- \(b^2 = 16\) (thus \(b = 4\))
- \(a^2 = a^2\) (we will find \(a\) later)
### Step 4: Use the tangent condition
The condition for the line \(y = mx + c\) to be a tangent to the hyperbola is given by:
\[ c^2 = a^2 m^2 - b^2 \]
Substituting the known values:
- \(c = 1\)
- \(m = 2\)
- \(b^2 = 16\)
We have:
\[
1^2 = a^2(2^2) - 16
\]
\[
1 = 4a^2 - 16
\]
\[
4a^2 = 17
\]
\[
a^2 = \frac{17}{4}
\]
Thus, \(a = \sqrt{\frac{17}{4}} = \frac{\sqrt{17}}{2}\).
### Step 5: Evaluate the options for right-angled triangle sides
We need to check which of the given options cannot be the sides of a right-angled triangle. The options are not provided, but we will denote them as \(x_1, x_2, x_3, x_4\).
For each option, we will identify the longest side and check the Pythagorean theorem \(c^2 = a^2 + b^2\).
#### Option A: \(2a, 4, 1\)
- Longest side = \(2a = 2 \cdot \frac{\sqrt{17}}{2} = \sqrt{17}\)
- Check:
\[
(\sqrt{17})^2 = 4^2 + 1^2 \Rightarrow 17 = 16 + 1 \Rightarrow 17 = 17 \quad \text{(True)}
\]
#### Option B: \(2a, 8, 1\)
- Longest side = \(8\)
- Check:
\[
8^2 = (2a)^2 + 1^2 \Rightarrow 64 = 4a^2 + 1
\]
Substituting \(a^2 = \frac{17}{4}\):
\[
64 = 4 \cdot \frac{17}{4} + 1 \Rightarrow 64 = 17 + 1 \Rightarrow 64 \neq 18 \quad \text{(False)}
\]
#### Option C: \(4, a, 1\)
- Longest side = \(4\)
- Check:
\[
4^2 = a^2 + 1^2 \Rightarrow 16 = \frac{17}{4} + 1
\]
\[
16 = \frac{17}{4} + \frac{4}{4} \Rightarrow 16 = \frac{21}{4} \quad \text{(False)}
\]
#### Option D: \(4, a, 2\)
- Longest side = \(4\)
- Check:
\[
4^2 = a^2 + 2^2 \Rightarrow 16 = \frac{17}{4} + 4
\]
\[
16 = \frac{17}{4} + \frac{16}{4} \Rightarrow 16 = \frac{33}{4} \quad \text{(False)}
\]
### Conclusion
From the evaluations:
- Option A can be the sides of a right-angled triangle.
- Options B, C, and D cannot be the sides of a right-angled triangle.
Thus, the answer to the question is that the sides which **cannot** form a right-angled triangle are options B, C, and D.
