Key Concepts and Formulas for Common Tangents

To solve this problem, which asks for a line simultaneously tangent to a parabola and a circle, we employ two fundamental concepts from coordinate geometry:

1. Equation of a Tangent to a Parabola ($y^2 = 4ax$): For a parabola of the form $y^2 = 4ax$ (vertex at the origin, opening to the right), the equation of a tangent line with a slope $m$ is given by: $y = mx + \frac{a}{m}$

   • Why this is useful: This formula allows us to express any tangent to the parabola in terms of a single variable, its slope $m$. This parametrization simplifies the process of finding a specific tangent that also satisfies other conditions.
   • Important Note: This formula is valid for $m \ne 0$. If $m=0$, the tangent would be $y=0$ (the x-axis), which is not a tangent to $y^2=4ax$ unless $a=0$. If $m$ were undefined (vertical tangent, $x=k$), the tangent to $y^2=4ax$ would be $x=0$, which is the axis of the parabola and only touches it at the vertex, but cannot be expressed in $y=mx+c$ form.

2. Condition for Tangency to a Circle ($x^2 + y^2 = r^2$): A line $Ax + By + C = 0$ is tangent to a circle $x^2 + y^2 = r^2$ (centered at the origin with radius $r$) if and only if the perpendicular distance from the center of the circle $(0,0)$ to the line is equal to the radius $r$. The distance formula for a point $(x_0, y_0)$ to a line $Ax+By+C=0$ is $\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$. Applying this for the center $(0,0)$: $\frac{|A(0) + B(0) + C|}{\sqrt{A^2 + B^2}} = r \quad \implies \quad \frac{|C|}{\sqrt{A^2 + B^2}} = r$

   • Why this is useful: This condition provides a direct algebraic way to check if a given line is tangent to a circle, without needing to find the point of tangency or solve quadratic equations for intersections. It's a powerful geometric property translated into an algebraic condition.

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Step-by-Step Solution

Step 1: Determine the general equation of a tangent to the parabola $y^2 = 4\sqrt{2}x$.

• Goal: Our first objective is to find the general form of any line that touches the given parabola. We will use the standard tangent formula.
• The given parabola equation is $y^2