Key Concepts and Formulas

• Section Formula: If a point $P(x, y)$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$, then $x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}$
• Distance Formula: The distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Equation of a Line: A line can be represented by an equation of the form $ax + by + c = 0$.

Step-by-Step Solution

Step 1: Parameterize the coordinates of points A and B.

The point $A$ lies on the line $x - y + 2 = 0$, so $x = y - 2$. Let $A = (t-2, t)$. The point $B$ lies on the line $y + 2 = 0$, so $y = -2$. Let $B = (s, -2)$.

Step 2: Use the section formula to find the coordinates of point P.

The point $P(x, y)$ divides the line segment $AB$ in the ratio $2:1$. Using the section formula: $x = \frac{2s + 1(t-2)}{2+1} = \frac{2s + t - 2}{3}$ $y = \frac{2(-2) + 1(t)}{2+1} = \frac{-4 + t}{3}$

Step 3: Express s and t in terms of x and y.

From the equations in Step 2, we can express $s$ and $t$ in terms of $x$ and $y$: $3x = 2s + t - 2 \implies 2s = 3x - t + 2$ $3y = -4 + t \implies t = 3y + 4$ Substituting $t$ into the equation for $2s$: $2s = 3x - (3y + 4) + 2 = 3x - 3y - 2$ $s = \frac{3x - 3y - 2}{2}$

Step 4: Use the distance formula to apply the condition AB = 8.

Since $AB = 8$, we have: $AB^2 = (s - (t-2))^2 + (-2 - t)^2 = 64$ Substituting the expressions for $s$ and $t$: $\left(\frac{3x - 3y - 2}{2} - (3y + 4 - 2)\right)^2 + (-2 - (3y + 4))^2 = 64$ $\left(\frac{3x - 3y - 2}{2} - (3y + 2)\right)^2 + (-6 - 3y)^2 = 64$ $\left(\frac{3x - 3y - 2 - 6y - 4}{2}\right)^2 + (3y + 6)^2 = 64$ $\left(\frac{3x - 9y - 6}{2}\right)^2 + (3y + 6)^2 = 64$ $\frac{9}{4}(x - 3y - 2)^2 + 9(y + 2)^2 = 64$ $9(x - 3y - 2)^2 + 36(y + 2)^2 = 256$ $9(x^2 + 9y^2 + 4 - 6xy - 4x + 12y) + 36(y^2 + 4y + 4) = 256$ $9x^2 + 81y^2 + 36 - 54xy - 36x + 108y + 36y^2 + 144y + 144 = 256$ $9x^2 + 117y^2 - 54xy - 36x + 252y + 180 = 256$ $9x^2 + 117y^2 - 54xy - 36x + 252y - 76 = 0$ Dividing by 9: $x^2 + 13y^2 - 6xy - 4x + 28y - \frac{76}{9} = 0$ $9(x^2 + 13y^2 - 6xy - 4x + 28y) - 76 = 0$

Step 5: Compare with the given equation and find the values of α, β, and γ.

Comparing the equation $9(x^2+\alpha y^2+\beta x y+\gamma x+28 y)-76=0$ with $9(x^2 + 13y^2 - 6xy - 4x + 28y) - 76 = 0$, we get: $\alpha = 13, \quad \beta = -6, \quad \gamma = -4$

Step 6: Calculate α - β - γ.

$\alpha - \beta - \gamma = 13 - (-6) - (-4) = 13 + 6 + 4 = 23$

Common Mistakes & Tips

• Be careful with signs when substituting and simplifying expressions.
• Double-check your algebraic manipulations to avoid errors.
• Remember to use the distance formula correctly.

Summary

We parameterized the endpoints of the rod using the given line equations. Then, we used the section formula to find the coordinates of point P in terms of the parameters. We expressed the parameters in terms of x and y, and finally, we used the distance formula and the given length of the rod to derive the locus of point P. By comparing the derived equation with the given equation, we found the values of α, β, and γ and calculated α - β - γ.

The final answer is \boxed{23}, which corresponds to option (D).