To find the common tangents to the circle \(x^2 + y^2 = 2a^2\) and the parabola \(y^2 = 8ax\), we need to analyze the equations and derive their common tangents.
- First, consider the standard equations:
- The circle is given by \(x^2 + y^2 = 2a^2\), which is centered at the origin with a radius of \(\sqrt{2}a\).
- The parabola is given by \(y^2 = 8ax\), which opens to the right and has vertex at the origin.
- Common tangents to a circle and a parabola can be found using the concept that they both lie at points where their respective distances from any point \((x, y)\) on the tangent line equate.
- We systematically derive the tangents:
- The general tangent to the parabola \(y^2 = 8ax\) can be given by the equation \(y = mx + \frac{2a}{m}\).
- The general tangent to the circle \(x^2 + y^2 = 2a^2\) can be written in the form: \(y = mx \pm \sqrt{2a^2(1 + m^2)}\).
- Equating these for common tangents involves equating coefficients, particularly of \(x\) and the constant terms.
- Additionally, substitute and simplify the expressions to find when they represent valid common tangents.
- After simplifying, we find the common tangents suitable with the structure \(y = \pm(x + 2a)\).
By verifying through balancing coefficients and forming valid tangents, the derived solution is the correct form of the common tangents present in the options given:
The correct answer is: \(y = \pm(x + 2a)\).