Step-by-step Solution:

To find the points on the hyperbola \( x^2 - y^2 = 3 \) where the tangent is parallel to the straight line \( 2x + y + 8 = 0 \), follow these steps: --- Step 1: Find the Slope of the Given Line The slope of the line \( 2x + y + 8 = 0 \) is: \[ m = -2 \] --- Step 2: Find the Slope of the Tangent to the Hyperbola The hyperbola is given by: \[ x^2 - y^2 = 3 \] Differentiate implicitly with respect to \( x \): \[ 2x - 2y \frac{dy}{dx} = 0 \] Solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{x}{y} \] The slope of the tangent to the hyperbola at any point \( (x, y) \) is \( \frac{x}{y} \). --- Step 3: Set the Slopes Equal For the tangent to be parallel to the given line, their slopes must be equal: \[ \frac{x}{y} = -2 \] Solve for \( x \): \[ x = -2y \] --- Step 4: Substitute into the Hyperbola Equation Substitute \( x = -2y \) into the hyperbola equation \( x^2 - y^2 = 3 \): \[ (-2y)^2 - y^2 = 3 \] \[ 4y^2 - y^2 = 3 \] \[ 3y^2 = 3 \] \[ y^2 = 1 \] \[ y = \pm 1 \] Now, find \( x \) using \( x = -2y \): - If \( y = 1 \), then \( x = -2 \). - If \( y = -1 \), then \( x = 2 \). --- Step 5: Identify the Points The points on the hyperbola where the tangent is parallel to the given line are: \[ (-2, 1) \quad \text{and} \quad (2, -1) \] --- Step 6: Match with the Given Options The points \( (-2, 1) \) and \( (2, -1) \) correspond to Option B. --- Final Answer: The correct option is: \[ \boxed{B} \]