If 2x – y + 1 = 0 is a tangent to the hyperbola x2/a2 – y2/16 = 1, then which of the following cannot be the sides of a right-angled triangle?

We are given that the equation of the line 2x - y + 1 = 0 is a tangent to the hyperbola x²/a² - y²/16 = 1, and we need to determine which of the following cannot be the sides of a right-angled triangle.

1. Equation of the hyperbola:
   • The given hyperbola is x²/a² - y²/16 = 1.
   
   
   • This is a standard form of a hyperbola with its transverse axis along the x-axis and conjugate axis along the y-axis.


2. Equation of the tangent line:
   • The equation of the tangent line is 2x - y + 1 = 0, or equivalently, y = 2x + 1.
   
   
   • The tangent to a hyperbola is given by the condition that the perpendicular distance from the center of the hyperbola (which is the origin in this case) to the line equals the semi-major axis (a) of the hyperbola.


3. Formula for perpendicular distance from the origin to a line:
   • The perpendicular distance from a point (x₀, y₀) to the line Ax + By + C = 0 is given by: d = |Ax₀ + By₀ + C| / √(A² + B²).
   
   
   • For the line 2x - y + 1 = 0, we have A = 2, B = -1, and C = 1, and the point is the origin (0, 0).
   
   
   • Thus, the perpendicular distance from the origin to the line is: d = |2(0) - (0) + 1| / √(2² + (-1)²) = |1| / √(4 + 1) = 1 / √5.


4. Condition for the tangent to touch the hyperbola:
   • For the line to be a tangent to the hyperbola, the perpendicular distance from the center (origin) to the line must be equal to the semi-major axis of the hyperbola (which is a). In other words, we must have: 1/√5 = a/4 (since the semi-major axis of the hyperbola is 4 due to y²/16 = 1, implying the semi-major axis is 4).
   
   
   • Solving for a, we get a = 4/√5.


5. Check the sides of a right-angled triangle:
   • For a right-angled triangle with sides a, b, c (where c is the hypotenuse), the Pythagorean theorem must hold: a² + b² = c².
   
   
   • Now, let's check the given options for the sides of a triangle. We need to identify which set of sides cannot form a right-angled triangle based on this condition.


6. Final Answer:
   • The correct answer will depend on the specific side lengths provided in the options, but by applying the Pythagorean theorem, you can rule out the set of sides that do not satisfy a² + b² = c².