This problem is a classic example of how concepts from different conic sections – parabolas and circles – are interconnected in coordinate geometry. The core idea is to first determine the equation of the tangent line to the parabola at a given point, and then use the condition that this same line is also tangent to a circle to find an unknown parameter of the circle.

---

Understanding the Key Concepts

Before we dive into the solution, let's refresh our memory on the essential formulas and conditions for tangents:

1. Tangent to a Parabola ($y^2 = 4ax$): When the point of tangency $(x_1, y_1)$ on the parabola $y^2 = 4ax$ is known, the equation of the tangent line is given by the formula: $yy_1 = 2a(x+x_1)$ This formula is derived from calculus (finding the derivative $\frac{dy}{dx}$ and using point-slope form) or from algebraic methods (discriminant of a quadratic equation). It's particularly efficient when the point of contact is provided.

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

---

Step-by-Step Solution

1. Determine the Equation of the Tangent Line L to the Parabola

Our first goal is to find the equation of the line L that is tangent to the parabola $y^2 = 8x$ at the point $(2, -4)$.

• Identify the Parabola's Standard Form: The given parabola is $y^2 = 8x$. We compare this to the standard form of a parabola opening to the right, $y^2 = 4ax$. By comparing coefficients, we have $4a = 8$. Solving for $a$, we get $a = 2$. Why this step? The parameter 'a' of the parabola is crucial for using the specific tangent formula $yy_1 = 2a(x+x_1)$.

• Identify the Point of Tangency: The problem explicitly states that the tangent line is drawn at the point $(2, -4)$. So, we have $(x_1, y_1) = (2, -4)$.

• Apply the Tangent Formula: Now we substitute the values of $a=2$, $x_1=2$, and $y_1=-4$ into the tangent formula $yy_1 = 2a(x+x_1)$: $y(-4) = 2(2)(x+2)$ Why this formula? This formula is specifically designed for finding the tangent at a given point on the parabola, making it the most direct method here.

• Simplify the Equation of Line L: Perform the multiplications and rearrange the terms to get the equation of the line in a standard form. $-4y = 4(x+2)$ $-4y = 4x + 8$ To simplify, divide the entire equation by 4: $-y = x + 2$ Rearrange this into the general form $Ax+By+C=0$, which is ideal for applying the perpendicular distance formula later: $x + y + 2 = 0 \quad \text{ (Equation of line L)}$

  Self-Check (using slope): For those familiar with finding the slope, the derivative of $y^2 = 8x$ is $2y \frac{dy}{dx} = 8$, so $\frac{dy}{dx} = \frac{4}{y}$. At $(2, -4)$, the slope $m = \frac{4}{-4} = -1$. The tangent line is $y - (-4) = -1(x - 2)$, which simplifies to $y+4 = -x+2$, or $x+y+2=0$. This confirms our result.

2. Apply the Tangency Condition for the Circle

Now we know the equation of line L is $x+y+2=0$. We are given that this line is also tangent to the circle $x^2 + y^2 = a$. Our goal is to find the value of this 'a'.

• Identify the Circle's Properties: The given circle is $x^2 + y^2 = a$. Why this step? We need to determine the center and radius of the circle to apply the tangency condition. Comparing this to the standard form of a circle centered at the origin, $x^2 + y^2 = r^2$:

  • The center of the circle is $(0,0)$.
  • The square of the radius is $r^2 = a$, so the radius is $r = \sqrt{a}$.
  • Important Note: This 'a' (the radius squared of the circle) is distinct from the 'a' parameter of the parabola we found earlier. It's a common trick in problems to use the same variable name for different quantities.

• Identify Parameters of Line L: The equation of line L is $x + y + 2 = 0$. Comparing this to the general form $Ax+By+C=0$, we have:

  • $A = 1$ (coefficient of $x$)
  • $B = 1$ (coefficient of $y$)
  • $C = 2$ (constant term)

• Apply the Perpendicular Distance Formula: For line L to be tangent to the circle, the perpendicular distance from the circle's center $(0,0)$ to line L must be equal to the circle's radius $\sqrt{a}$. Using the simplified perpendicular distance formula for a point $(0,0)$ to a line $Ax+By+C=0$: $d = \frac{|C|}{\sqrt{A^2+B^2}}$ Substitute the values $A=1$, $B=1$, and $C=2$: $d = \frac{|2|}{\sqrt{1^2 + 1^2}}$ Why this formula? This is the fundamental condition for a line to be tangent to a circle: the distance from the center to the line must be exactly the radius.

• Calculate the Distance: $d = \frac{2}{\sqrt{1+1}}$ $d = \frac{2}{\sqrt{2}}$ Rationalize the denominator: $d = \frac{2\sqrt{2}}{2} = \sqrt{2}$

• Equate Distance to Radius and Solve for 'a': Now, we set the calculated perpendicular distance $d$ equal to the circle's radius $r$: $d = r$ $\sqrt{2} = \sqrt{a}$ To find the value of 'a', we square both sides of the equation: $(\sqrt{2})^2 = (\sqrt{a})^2$ $2 = a$

Thus, the value of 'a' for the circle is 2.

---

Important Tips and Common Pitfalls

• Variable Distinction: Always be mindful when a problem uses the same variable name ('a' in this case) for different quantities. Clearly define what each 'a' represents in your solution to avoid confusion.
• Choosing the Correct Tangent Formula: For parabolas, if the point of tangency $(x_1, y_1)$ is given, the formula $yy_1 = 2a(x+x_1)$ is almost always the most efficient. If only the slope $m$ is given, then $y=mx+\frac{a}{m}$ would be preferred.
• General Form of a Line: When using the perpendicular distance formula, ensure your line equation is in the general form $Ax+By+C=0$. This prevents errors in identifying $A, B,$ and $C$.
• Absolute Value: Don't forget the absolute value in the numerator of the perpendicular distance formula. Distance is always non-negative.
• Radius vs. Radius Squared: Remember that for $x^2+y^2=a$, 'a' is $r^2$, so the radius is $\sqrt{a}$.

---

Summary and Key Takeaway

This problem effectively tests your understanding of tangent properties for both parabolas and circles. We successfully determined the tangent line to the parabola using a specific point-of-contact formula, and then applied the geometric condition of tangency (distance from center equals radius) to the circle to find the unknown parameter 'a'. The final value of 'a' for the circle is $\mathbf{2}$. This problem emphasizes the importance of knowing and correctly applying standard formulas and geometric conditions in coordinate geometry.