Introduction
To find the equation of the common tangent to the circles and parabolas given by the equations x² + y² = 2a² and y² = 8aX, we will derive the tangent equations for both curves and equate them.
Step 1: Circle Equation
- The equation x² + y² = 2a² represents a circle centered at the origin with a radius of √(2a²) = a√2.
- The general equation of the tangent to this circle can be expressed as:
y = mx ± √(2a²(1 + m²))
Step 2: Parabola Equation
- The equation y² = 8aX represents a parabola that opens to the right.
- The equation of the tangent to this parabola can be written as:
y = mx + a/m
Step 3: Equating Tangents
- For the tangents to be common, both equations must be equal in terms of y.
- Therefore, we equate:
mx + a/m = mx ± √(2a²(1 + m²))
Step 4: Solving for m
- By rearranging and simplifying, we can derive a quadratic equation in terms of m.
- This will yield the slopes of the tangents that are common to both the circle and the parabola.
Final Equation
- After solving the quadratic equation, substitute the value of m back into either tangent equation to get the common tangent line's equation.
Conclusion
The common tangent would result in the equation of the form y = mx + c, where m is the slope and c is derived from the constants in the equations of the curves. This method provides a systematic approach to finding common tangents between different conic sections.