1. Introduction to Key Concepts

This problem involves finding a common tangent line to a parabola and an ellipse. We will utilize two fundamental concepts from coordinate geometry to solve it:

• Equation of a Tangent to a Parabola at a Given Point ($T=0$ Method): If a point $(x_1, y_1)$ lies on a parabola, the equation of the tangent at that point can be efficiently found using the $T=0$ method. This method involves a specific substitution rule for the terms in the parabola's equation:

  • Replace $y^2$ with $yy_1$
  • Replace $x^2$ with $xx_1$
  • Replace $x$ with $\frac{x+x_1}{2}$
  • Replace $y$ with $\frac{y+y_1}{2}$
  • Constant terms remain unchanged.

• Condition for Tangency of a Line to an Ellipse: A straight line given by the equation $y = mx + c$ is tangent to the standard ellipse $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ if and only if the following condition is satisfied: $c^2 = A^2m^2 + B^2$

Our strategy will be to first determine the equation of the tangent line to the parabola at the given point. We will then express this line in the slope-intercept form ($y=mx+c$) to identify its slope ($m$) and y-intercept ($c$). Finally, we will use these values along with the tangency condition for the ellipse to solve for the unknown parameter $b$.

2. Step 1: Finding the Equation of Tangent Line L to the Parabola

Problem Statement: We are given the parabola $y^2 = 4x - 20$ and a point $(6, 2)$ on it. Why this step: The first crucial step is to determine the precise equation of the tangent line $L$. Since we are given a point that lies on the parabola, the $T=0$ method provides a direct and efficient way to find this tangent equation.

Before applying the $T=0$ method, it's good practice to verify that the given point $(6, 2)$ indeed lies on the parabola. Substitute $x=6$ and $y=2$ into the parabola equation: $(2)^2 = 4(6) - 20$ $4 = 24 - 20$ $4 = 4$ Since the equation holds true, the point $(6, 2)$ lies on the parabola.

Now, we apply the $T=0$ method to find the tangent at $(x_1, y_1) = (6, 2)$. The parabola equation is $y^2 = 4x - 20$. We can rearrange it to $y^2 - 4x + 20 = 0$ for easier application of the $T=0$ rule.

Applying the $T=0$ transformations:

• $y^2$ becomes $yy_1 = y(2) = 2y$
• $-4x$ becomes $-4\left(\frac{x+x_1}{2}\right) = -4\left(\frac{x+6}{2}\right) = -2(x+6)$
• The constant term $+20$ remains $+20$

So, the equation of the tangent line $L$ is: $2y - 2(x+6) + 20 = 0$