Step 1: Find the Focus of the Parabola
The standard equation of a rightward-opening parabola is $$y^2 = 4ax$$.
Comparing this to our given equation:
$$y^2 = 12x \implies 4a = 12 \implies a = 3$$
The focus of the parabola is located at $$(a, 0)$$, which is $$(3, 0)$$.
Step 2: Set up the Equation of the Focal Chord
Any line passing through the focus $$(3, 0)$$ with a slope $$m$$ can be written using the point-slope form:
$$y - 0 = m(x - 3)$$ $$mx - y - 3m = 0$$
Step 3: Apply the Condition of Tangency to the Circle
The given equation of the circle is:
$$(x - 5)^2 + y^2 = 2$$
From this standard form, we can identify:
- Center ($$C$$): $$(5, 0)$$
- Radius ($$R$$): $$\sqrt{2}$$
For the line to be a tangent to the circle, the perpendicular distance from the center $$(5, 0)$$ to the line must equal the radius $$R$$:
$$\frac{|m(5) - (0) - 3m|}{\sqrt{m^2 + (-1)^2}} = \sqrt{2}$$
Simplify the expression inside the absolute value:
$$\frac{|2m|}{\sqrt{m^2 + 1}} = \sqrt{2}$$
Step 4: Solve for the Slopes ($$m$$)
Square both sides of the equation to eliminate the modulus and radical signs:
$$\frac{4m^2}{m^2 + 1} = 2$$
Cross-multiply and simplify:
$$4m^2 = 2(m^2 + 1)$$ $$4m^2 = 2m^2 + 2$$ $$2m^2 = 2 \implies m^2 = 1$$
This gives two possible slopes:
Step 5: Calculate the Required Value
Substitute the values of $$m_1^2$$ and $$m_2^2$$ into the target expression:
$$4(m_1^2 + m_2^2) = 4(1 + 1) = 4(2) = 8$$
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